PART 2: DETERMINANTS, GENERAL VECTOR SPACES, AND MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS 3.1: THE DETERMINANT OF A MATRIX Learning Objectives 1. Find the determinant of a 2 x 2 matrix 2. Find the minors and cofactors of a matrix 3. Use expansion by cofactors to find the determinant of a matrix 4. Find the determinant of a triangular matrix 5. Use elementary row operations to evaluate a determinant 6. Use elementary column operations to evaluate a determinant 7. Recognize conditions that yield zero determinants
Every ______matrix can be associated with a real number called its ______.
Historically, the use of determinants arose from the recognition of special ______that occur in the ______of systems of linear equations.
DEFINITION OF THE DETERMINANT OF A 2 x 2 MATRIX
The ______of the matrix
a11 a 12 A a21 a 22 is given by det A ______.
**Note: In this text, ______and ______are used interchangeably to represent the determinant of a matrix. In this context, the vertical bars are used to represent the ______of a matrix as opposed to the ______value. Example 1: Find det Aand detB . . 1 4 21 3 b. a. A B 11 7 6 10
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DEFINITION OF MINORS AND COFACTORS OF A MATRIX
If A is a ______matrix, then the ______of the element _____ is the determinant of the matrix obtained by deleting the ____ row and the _____ column of A .The ______
_____ is given by Cij ______.
Example 2: Find the minor and cofactor of a12 .and b23 .
a11 a 12 a 13 2 1 4 A a a a B 0 1 3 21 22 23 a31 a 32 a 33 3 2 1
DEFINITION OF THE DETERMINANT OF A SQUARE MATRIX
If A is a ______matrix of order n 2 , then the ______of the A is the _____ of the entries in the first row of A multiplied by their respective ______. That is, n det A A a1j C 1 j ______. j1
Example 3: Confirm that, for 2x2 matrices, this definition yields A a11 a 22 a 21 a 12 .
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Example 4: Find B . 2 1 4 B 0 1 3 3 2 1
THEOREM 3.1: DETERMINANT OF A MATRIX PRODUCT If A be a square matrix of order n . Then the determinant of A is given by n det A A aij C ij ______(ith row expansion) j1 n det A A aij C ij ______(jth column expansion) i1
Is there an easier way to complete the previous example?
2 1 4 B 0 1 3 3 2 1
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Alternative Method to evaluate the determinant of a 3 x 3 matrix: Copy the first and second columns of the matrix to form fourth and fifth columns. Then obtain the determinant by adding (or subtracting) the products of the six diagonals.
2 1 4 B 0 1 3 3 2 1
Example 5: Find det A. 1 0 0 0 3 7 0 0 A 6 1 2 0 3 5 8 7
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What did you notice?
THEOREM 3.2: DETERMINANT OF A TRIANGULAR MATRIX If A is a triangular matrix of order n , then its determinant is the ______of the ______on the ______. That is, det AA ______.
Example 6: Find the values of , for which the determinant is zero. 1 1 4 3
Consider the following two matrices: 1 2 1 1 2 1 A 3 4 1 B 0 10 2 1 0 1 0 0 8 Find the determinant of each matrix.
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What did you find out?
Take a closer look at the two matrices. Do you notice anything?
THEOREM 3.3: ELEMENTARY ROW OPERATIONS AND DETERMINANTS Let A and B be square matrices. 1. When B is obtained from A by ______two ______of A , ______.
2. When B is obtained from A by ______a ______of a row of A to another row
of A , ______.
3. When B is obtained from A by ______a row of A by a ______
______c , ______.
NOTE: Theorem 3.3 remains valid when the word “column” replaces the word “row”. Operations performed on columns are called elementary column operations.
Example 7: Determine which property of determinants the equation illustrates.
1 1 3 3 1 1 a. 4 12 7 7 12 4 3 3 8 8 3 3
2 4 2 1 2 1 b. 6 10 2 8 3 5 1 8 4 6 4 2 3
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Example 8: Use elementary row or column operations to find the determinant of the matrix. 3 8 7 A 0 5 4 6 1 6
THEOREM 3.4: CONDITIONS THAT YIELD A ZERO DETERMINANT If A is a square matrix, and any one of the following conditions is true, then det A 0 . 1. An entire ______(or ______) consists of ______.
2. Two ______(or ______) are ______.
3. One ______(or ______) is a ______of another ______(or______).
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Example 9: Prove the property. 1 a 1 1 1 1 1 1 1b 1 abc 1 , a 0, b 0, c 0 . a b c 1 1 1 c
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3.2: PROPERTIES OF DETERMINANTS Learning Objectives 1. Find the determinant of a matrix product and a scalar multiple of a matrix 2. Find the determinant of an inverse matrix and recognize equivalent conditions for a nonsingular matrix 3. Find the determinant of the transpose of a matrix 4. Use Cramer’s Rule to solve a system of linear equations 5. Use determinants to find area, volume, and equations of lines and planes Example 1: Find A , B , AB , AB , AB and AB . 2 0 1 2 1 4 B 0 1 3 A 1 1 2 3 1 0 3 2 1
THEOREM 3.5: DETERMINANT OF A MATRIX PRODUCT If A and B are square matrices of order n , then
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Example 2: Find 3A and 3B . 2 1 4 1 1 A B 0 1 3 3 10 3 2 1
THEOREM 3.6: DETERMINANT OF A SCALAR MULTIPLE OF A MATRIX If A is a square matrix of order n and c is a scalar, then the determinant of cA is
Proof:
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Example 3: Find A1 , A , B 1 , and B . 3 6 5 2 A B 2 4 11 7
THEOREM 3.7: DETERMINANT OF AN INVERTIBLE MATRIX A square matrix A is invertible (nonsingular) if and only if
Example 4: Find AA and 1 . 3 3 A 2 1
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THEOREM 3.8: DETERMINANT OF AN INVERSE MATRIX If A is an n n invertible matrix, then
Proof:
EQUIVALENT CONDITIONS FOR A NONSINGULAR MATRIX If A is an n n matrix, then the following statements are equivalent. 1. A is ______.
2. Ax b has a ______solution for every ______column matrix.
3. Ax 0 has only the ______solution.
4. A is ______to ______.
5. A can be written as the product of ______matrices.
6. ______.
Example 5: Determine if the system of linear equations has a unique solution.
x1 x 2 x 3 4
2x1 x 2 x 3 6
3x1 2 x 2 2 x 3 0
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Example 6: Find AA and T . 7 12 A 2 2
THEOREM 3.9: DETERMINANT OF A TRANSPOSE If A is a square matrix, then
Example 7: Solve the system of linear equations. Assume that a11 a 22 a 21 a 12 0 . a11 x 1 a 12 x 2 b 1 a21 x 1 a 22 x 2 b 2
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THEOREM 3.10: CRAMER’S RULE
If a system of n linear equations in n variables has a coefficient matrix A with a nonzero determinant A , then the solution of the system is
Where the ith column of Ai is the column of constants in the system of equations. Example 8: If possible, use Cramer’s Rule to solve the system.
a. x 2 x 7 1 2 2x1 4 x 2 11
b.
8x1 7 x 2 10 x 3 151
12x1 3 x 2 5 x 3 86
15x1 9 x 2 2 x 3 187
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AREA OF A TRIANGLE IN THE xy-PLANE
The area of a triangle with vertices x1, y 1 , x 2 , y 2 , and x 3 , y 3 is
where the sign ( ) is chosen to give positive area.
Proof:
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Example 9: Find the area of the triangle whose vertices are 1, 1 , 3, 5 , and 0, 2 .
TEST FOR COLLINEAR POINTS IN THE xy-PLANE
Three points x1, y 1 , x 2 , y 2 , and x 3 , y 3 are collinear if and only if
TWO-POINT FORM OF THE EQUATION OF A LINE
An equation of the line passing through the distinct points x1, y 1 and x2, y 2 is given by
Example 10: Find an equation of the line passing through the points 4, 7 and 2, 4 .
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VOLUME OF A TETRAHEDRON
The volume of a tetrahedron with vertices x1,, y 1 z 1 , x2,, y 2 z 2 , x3,, y 3 z 3 , and x4,, y 4 z 4 is
where the sign ( ) is chosen to give positive volume.
Example 11: Find the volume of the tetrahedron with vertices 1,1,1 , 0,0,0 , 2,1, 1 , and 1,1,2 .
TEST FOR COPLANAR POINTS IN SPACE
Four points, x1,, y 1 z 1 , x2,, y 2 z 2 , x3,, y 3 z 3 , and x4 ,, y 4 z 4 are coplanar if and only if
where the sign ( ) is chosen to give positive volume.
THREE-POINT FORM OF THE EQUATION OF A LINE
An equation of the plane passing through the distinct points x1,, y 1 z 1 , x2,, y 2 z 2 ,and x3,, y 3 z 3 is given by
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3.3: GENERAL VECTOR SPACES Learning Objectives: 1. Determine whether a set of vectors is a vector space 2. Determine if a subset of a known vector space V is a subspace of V 3. Write a vector as a linear combination of other vectors n 4. Recognize bases in the vector spaces R , Pn , and M m, n 5. Determine whether a set S of vectors in a vector space V is a basis for V 6. Find the dimension of a vector space DEFINITION OF A VECTOR SPACE Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the listed axioms are satisfied for every u , v , and w in V and every scalar (real number) c and d , then V is called a vector space. Addition 1. u v is in V . ______under addition
2. u v ______property
3. u v w ______property
4. V has a ______vector ____ such that for every ___ in V , ______. additive ______
5. For every __ in V , there is a vector in V denoted by __ such that ______. additive ______
Scalar Multiplication
6. cu is in____. ______under scalar mult.
7. cu v ______property
8. c d u ______property
9. c du ______property
10. 1u ______identity
THEOREM 3.11: PROPERTIES OF SCALAR MULTIPLICATION Let v be any element of a vector space V , and let c be any scalar. Then the following properties are true. 1. 0v ______3. If ______, then ______or ______.
2. c0 ______4. 1 v ______
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Example 1: Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify at least one of the ten vector space axioms that fails. a b a. The set of all 2 x 2 matrices of the form S :,,, a b c d R . c 1
b. The set of all 2 x 2 nonsingular matrices with the standard operations.
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IMPORTANT VECTOR SPACES CONTINUED
______= the set of all ______defined on the real ______line.
______= the set of all ______defined on a ______
______.
_____ = the set of all ______.
_____ = the set of all ______of degree ______.
______= the set of all ______matrices.
______= the set of all ______matrices.
Example 2: Describe the zero vector (the additive identity) of the vector space.
a. C , b. M1,4
Example 3: Describe the additive inverse of a vector in the vector space.
a. C , b. M1,4
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Example 4: Determine whether the set of continuous functions, C , is a vector space.
1. Closure under addition.
2. Commutativity under addition.
3. Associativity under addition.
4. Additive identity.
5. Additive inverse.
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6. Closure under scalar multiplication.
7. Distributivity under scalar multiplication (2 vectors and 1 scalar).
8. Distributivity under scalar multiplication (2 scalars and 1 vector).
9. Associativity under scalar multiplication.
10. Scalar multiplicative identity.
Conclusion?
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Example 5: Determine whether the set W is a subspace of the vector space V with the standard operations of addition and scalar multiplication. a. WC: 1,1 V : The set of all functions that are differentiable on 1,1
b. W : The set of all negative functions: f x 0 . VC:,
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c. W : The set of all odd functions: f x f x . VC:,
d. W : The set of all n x n diagonal matrices. V:: Mn, n n Z
e. W : The set of all n x n matrices whose trace is nonzero. V:: Mn, n n Z
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f. W: ax b : a , b R , a 0 VC:, .
T g. W: a 0 a : a R , a 0 V::, Mm, n m n Z
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Example 6: For the matrices 2 3 0 5 A and B 4 1 1 2 in M 2,2 , determine whether the given matrix is a linear combination of A and B . 6 19 10 7
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Example 7: Determine whether the set of vectors in P2 is linearly independent or linearly dependent. S x2, x 2 1
Example 8: Determine whether the set of vectors in M 2,2 is linearly independent or linearly dependent. 2 0 4 1 8 3 S ,, 3 1 0 5 6 17
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Example 9: Write the standard basis for the vector space.
a. M 5,2
b. P3
Example 10: Determine whether S is a basis for the indicated vector space. 2 3 3 2 S4 t t ,5 t ,3 t 5, 2 t 3 t for P3
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Example 11: Find a basis for the vector space of all 3 x 3 symmetric matrices. What is the dimension of this vector space?
1 Example 11: Let T be the linear transformation from P2 into R given by the integral T p p x dx . 0 Find the preimage of 1. That is, find the polynomial function(s) of degree 2 or less such that T p 1 .
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3.4: RANK/NULLITY OF A MATRIX, SYSTEMS OF LINEAR EQUATIONS. AND COORDINATE VECTORS Learning Objectives: 1. Find a basis for the row space, a basis for the column space, and the rank of a matrix 2. Find the nullspace of a matrix 3. Find a coordinate matrix relative to a basis in Rn 4. Find the transition matrix from the basis B to the basis B in Rn 5. Represent coordinates in general n-dimensional spaces
Let’s do our math stretches! Consider the following matrix.
1 3 1 5 A 7 1 13 6 The row vectors of A are: The column vectors of A are:
DEFINITION OF ROW SPACE AND COLUMN SPACE OF A MATRIX Let A be an m n matrix. The ______space of A is the ______of R n ______by the ______vectors of A .
The ______space of A is the subspace of R n ______by the ______vectors of A .
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Recall that two matrices are row-equivalent when one can be obtained from the other by ______operations.
THEOREM 3.12: ROW-EQUIVALENT MATRICES HAVE THE SAME ROW SPACE If an m n matrix A is row-equivalent to an m n matrix B , then the row space of A is equal to the row space of B . Proof:
THEOREM 3.12: BASIS FOR THE ROW SPACE OF A MATRIX If a matrix A is row-equivalent to a matrix B in row-echelon form, then the nonzero row vectors of B form a
______for the row space of A .
To find a basis for the row space of a matrix: ______reduce the matrix. The ______rows in the
______matrix are a ______for the row space of the matrix. Your answer should be in the form of a ______of ______vectors.
To find a basis for the column space of a matrix:
Method 1: Use the steps above on the transpose of the matrix. Your answer should be in the form of a ______of
______vectors.
Method 2: Use reduced form of the original matrix to find the columns which contain the ______(leading
______). Use the corresponding columns from the ______matrix for a basis. Your answer should be in the form of a ______of ______vectors.
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Example 1: Find a basis for the row space and column space of the following matrix: 2 3 1 A 5 10 6 8 7 5
Example 2: Find a basis for the row space and column space of the following matrix: 4 20 31 A 6 5 6 2 11 16
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THEOREM 3.13: ROW AND COLUMN SPACES HAVE EQUAL DIMENSIONS
If A is an m n matrix, then the row space and the column space of A have the same ______.
DEFINITION OF THE RANK OF A MATRIX
The ______of the ______(or ______) space of a matrix A is called the
______of A and is denoted by ______.
Example 3: Find the rank of the matrix from a. Example 1 b. Example 2
THEOREM 3.14: SOLUTIONS OF A HOMOGENEOUS SYSTEM
If A is an m n matrix, then the set of all solutions of the homogeneous system of linear equations
______is a ______of _____ called the ______of _____ and is denoted
______. So,
The ______of the nullspace of A is called the ______of ____.
Proof:
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Example 4: Find the nullspace of the following matrix A , and determine the nullity of A . 1 4 2 1 A 0 1 1 1 2 8 4 2
THEOREM 3.15: DIMENSION OF THE SOLUTION SPACE
If A is an m n matrix of rank _____, then the ______of the solution space of
______is ______. That is,
Example 5: consider the following homogeneous system of linear equations: x y 0
x y 0 a. Find a basis for the solution space.
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b. Find the dimension of the solution space.
c. Find the solution of a consistent system Ax b in the form xp x h
THEOREM 3.16: SOLUTIONS OF A NONHOMOGENEOUS LINEAR SYSTEM
If x p is a particular solution of the nonhomogeneous system Ax b , then every solution of this system can be written in the form ______where xh is a solution of the corresponding homogeneous system______.
Proof:
THEOREM 3.17: SOLUTIONS OF A SYSTEM OF LINEAR EQUATIONS
The system ______is consistent if and only if ____ is in the column space of ______.
Proof:
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Example 7: consider the following nonhomogeneous system of linear equations: 2x 4 y 5 z 8 7x 14 y 4 z 28 3x 6 y z 12 Determine whether Ax b is consistent.
If the system is consistent, write the solution in the form x xp x h , where x p is a particular solution of
Ax b and xh is a solution of Ax 0 .
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COORDINATE REPRESENTATION RELATIVE TO A BASIS _____ Let B v1,,..., v 2 v n be an ordered basis for a vector space V , and let x be a vector in V such that
The scalars c1, c 2 ,..., cn are called the ______of ____ relative to the ______. The
______matrix (or coordinate ______) of _____ relative to _____ is the ______matrix in
____ whose ______are the coordinates of _____.
Note: In ____, column notation is used for the coordinate matrix. For the vector ______, the
______are the coordinates of ____ (relative to the ______for _____. So you have
Example 8: Find the coordinate matrix of x in Rn relative to the standard basis. x 1, 3,0
Example 9: Given the coordinate matrix of x relative to a (nonstandard) basis B for Rn , find the coordinate matrix of x relative to the standard basis. B 4,0,7,3 , 0,5, 1, 1 , 3,4,2,1 , 0,1,5,0 2 3 x B 4 1
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Example 10: Find coordinate matrix of x in Rn relative to the basis B. B 6,7 , 4, 3 , x 26,32
The matrix _____ is called the ______from ____ to _____, where ______is the coordinate matrix of ____ relative to _____, and ______is the coordinate matrix of _____ relative to _____.
Multiplication by the transition matrix ____ changes a coordinate matrix relative to _____ into a coordinate matrix relative to _____.
Change of basis from _____ to ____:
Change of basis from _____ to ____:
The change of basis problem in example 3 can be represented by the matrix equation:
THEOREM 3.18: THE INVERSE OF A TRANSITION MATRIX If P is the transition matrix from a basis B to a basis B in R n , then ____ is invertible and the transition matrix from ____ to ____ is given by _____.
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LEMMA
Let B v1, v 2 ,..., vn and B u1, u 2 ,..., un be two bases for a vector space V . If v1c 11 u 1 c 21 u 2 cn 1 u n v2c 12 u 1 c 22 u 2 cn 2 u n
vnc1 n u 1 c 2 n u 2 c nn u n then the transition matrix from ____ to ____ is
c11 c 1n Q cn1 c nn
THEOREM 3.19: TRANSITION MATRIX FROM B TO B’ n Let B v1,,..., v 2 v n and B u1, u 2 ,..., un be two bases for R . Then the transition matrix ______from _____ to _____ can be found using Gauss-Jordan elimination on the n 2 n matrix BB as follows.
Example 11: Find the transition matrix from B to B. BB1,1,1,0 , 1,0,0,1
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Example 12: Find the coordinate matrix of p relative to the standard basis for P3 . p3 x2 114 x 13
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3.5: THE KERNEL, RANGE, AND MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS, AND SIMILAR MATRICES Learning Objectives: 1. Find the kernel of a linear transformation 2. Find a basis for the range, the rank, and the nullity of a linear transformation 3. Determine whether a linear transformation is one-to-one or onto 4. Determine whether vector spaces are isomorphic 5. Find the standard matrix for a linear transformation 6. Find the standard matrix for the composition of linear transformations and find the inverse of an invertible linear transformation 7. Find the matrix for a linear transformation relative to a nonstandard basis 8. Find and use a matrix for a linear transformation 9. Show that two matrices are similar and use the properties of similar matrices
THE KERNEL OF A LINEAR TRANSFORMATION We know from an earlier theorem that for any linear transformation ______, the zero vector in ____ maps to the ______vector in _____. That is, ______. In this section, we will consider whether there are other vectors ____ such that ______. The collection of all such ______is called the ______of _____. Note that the zero vector is denoted by the symbol_____ in both _____ and
_____, even though these two zero vectors are often different.
DEFINITION OF KERNEL OF A LINEAR TRANSFORMATION Let TVW: be a linear transformation. Then the set of all vectors v in V that satisfy ______is called the ______of T and is denoted by ______.
Example 1: Find the kernel of the linear transformation. a. T: R3 R 3 , T x , y , z x , 0, z
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2 3 2 b. TP:3 PTa 2 , 0 axax 1 2 ax 3 a 1 2 ax 2 3 ax 3
c.
TPR:,2 1 T p p x dx 0
THEOREM 3.20: THE KERNEL IS A SUBSPACE OF V The kernel of a linear transformation TVW: is a subspace of the domain V .
Proof:
THEOREM 3.20: COROLLARY Let TRR: n m be the linear transformation given by TAx x . Then the kernel of T is equal to the solution space of ______.
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THEOREM 3.21: THE RANGE OF T IS A SUBSPACE OF W The range of a linear transformation TVW: is a subspace of W .
THEOREM 3.21: COROLLARY Let TRR: n m be the linear transformation given by TAx x . Then the column space of ______is equal to the ______of _____.
Example 2: Let TAv v represent the linear transformation T . Find a basis for the kernel of T and the range of T . 1 1 A 1 2 0 1
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DEFINITION OF RANK AND NULLITY OF A LINEAR TRANSFORMATION
Let TVW: be a linear transformation. The dimension of the kernel of T is called the
______of T and is denoted by ______. The dimension of the range of T is called the ______of T and is denoted by ______.
THEOREM 3.22: SUM OF RANK AND NULLITY Let TVW: be a linear transformation from an n-dimensional vector space V into a vector space W . Then the ______of the ______of the ______and ______is equal to the dimension of the ______. That is,
Proof:
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Example 3: Define the linear transformation T by TAx x . Find ker T , nullT , rangeT , and rank T . 3 2 6 1 15 A 4 3 8 10 14 2 3 4 4 20
Example 4: Let TRR: 3 3 be a linear transformation. Use the given information to find the nullity of T and give a geometric description of the kernel and range of T . T is the reflection through the yz-coordinate plane: T x,,,, y z x y z
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ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS If the ______vector is the only vector ____ such that ______, then ____ is
______. A function ______is called one-to-one when the
______of every _____ in the range consists of a ______vector. This is equivalent to saying that ____ is one-to-one if and only if, for all ____ and ____ in _____, ______implies that ______.
THEOREM 3.23: ONE-TO-ONE LINEAR TRANSFORMATIONS
Let TVW: be a linear transformation. Then T is one-to-one if and only if ______.
Proof:
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THEOREM 3.24: LINEAR TRANSFORMATIONS Let TVW: be a linear transformation, where W is finite dimensional. Then T is onto if and only if the ______of T is equal to the ______of W .
Proof:
THEOREM 3.25: ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS
Let TVW: be a linear transformation with vector spaces V and W , ______of dimension n. Then
T is one-to-one if and only if it is ______.
Example 5: Determine whether the linear transformation is one-to-one, onto, or neither. T:,,, R2 R 2 T x y x y y x
DEFINITION: ISOMORPHISM
A linear transformation TVW: that is ______and ______is called an
______. Moreover, if V and W are vector spaces such that there exists an isomorphism from V to W , then V and W are said to be ______to each other.
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THEOREM 3.26: ISOMORPHIC SPACES AND DIMENSION
Two finite dimensional vector spaces V and W are ______if and only if they are of the same ______.
Example 6: Determine a relationship among m, n, j, and k such that M m, n is isomorphic to M j, k .
WHICH FORMAT IS BETTER? WHY? 3 3 Consider TR: RTxxx ,,, 123 4 xx 123 5,2 x xx 12 6,3 xx 323 x and
4 1 5 x1 Tx A x 2 1 6 x 2 0 1 3 x3 What do you think?
The key to representing a linear transformation ______by a matrix is to determine how it acts on a
______for _____. Once you know the ______of every vector in the ______, you can use the properties of linear transformations to determine ______for any ___ in ____.
n n Do you remember the standard basis for R ? Write this standard basis for R in column vector notation.
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B e1,,..., e 2 en
THEOREM 3.26: STANDARD MATRIX FOR A LINEAR TRANSFORMATION n m n Let TRR: be a linear transformation such that, for the standard basis vectors ei of R ,
a11 a 12 a 1n a a a 21 22 2n TTTe1 , e 2 , , en , am1 a m 2 a mn then the m n matrix whose n columns correspond to T ei
a11 a 1n A am1 a mn is such that TAv v for every v in R n . A is called the standard matrix for T .
Example 5: Find the standard matrix for the linear transformation T. T x, y 4 x y , 0, 2 x 3 y
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Example 2: Use the standard matrix for the linear transformation Tto find the image of the vector v. T x, y x y , x y ,2,2, x y v 3,3
Example 6: Consider the following linear transformation T: T is the reflection through the yz-coordinate plane in R3 : T x ,, y z x ,,, y z v 2,3,4 . a. Find the standard matrix A for the following linear transformation T .
b. Use A to find the image of the vector v .
c. Sketch the graph of v and its image.
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THEOREM 3.27: COMPOSITION OF LINEAR TRANSFORMATIONS n m m p Let TRR1 : and TRR2 : be linear transformations with standard matrices A1 and A2 , respectively. n p The composition TRR: , defined by TTTv 2 1 v , is a linear transformation. Moreover, the
standard matrix A for T is given by the matrix product AAA 2 1 .
Proof:
Example 7: Find the standard matrices A and A for TTT 2 1 and TTT 1 2 . T: R2 R 3 , T x , y x , y , y 1 1 3 2 T2: R R , T 2 x , y , z y , z
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DEFINITION OF INVERSE LINEAR TRANSFORMATION n n n n n If TRR1 : and TRR2 : are linear transformations such that for every v in R ,
then T2 is called the ______of T1 , and T1 is said to be ______.
**Not every ______transformation has an ______. If ____ is ______, however, the inverse is ______and is denoted by ______.
THEOREM 3.28 Let is TRR: n n be a linear transformation with a standard matrix A . Then the following conditions are equivalent. T is ______.
T is an ______.
A is ______.
If T is invertible with standard matrix A , then the standard matrix for _____ is ______.
Example 8: Determine whether the linear transformation T x,, y x y x y is invertible. If it is, find its inverse.
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THEOREM 3.29: TRANSFORMATION MATRIX FOR NONSTANDARD BASES Let V and W be finite-dimensional vector spaces with bases B and B , respectively, where
B v1,,..., v 2 v n . If TVW: is a linear transformation such that
a11 a 12 a 1n a21 a 22 a 2n TTTv , v , , v , 1 BBB 2 n am1 a m 2 a mn then the m n matrix whose n columns correspond to T v 1 B
a11 a 1n A am1 a mn is such that ______for every ____ in ____.
Example 9: Find T v by using (a) the standard matrix, and (b) the matrix relative to B and B . TR:3 RTxyz 2 , ,, xyyz , , v 1,2,3,
BB1,1,1 , 1,1,0 , 0,1,1 , 1,2 ,1,1
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Example 10: Let B e2x,, xe 2 x x 2 e 2 x be a basis for a subspace of W of the space of continuous functions, and let Dx be the differential operator on W . Find the matrix for Dx relative to the basis B .
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A classical problem in linear algebra is determining whether it is possible to find a basis _____ such that the matrix for _____ relative to _____ is ______.
1. Matrix for T relative to B : ______2. Matrix for T relative to B : ______3. Transition matrix from B to B : ______4. Transition matrix from B to B : ______
Example 11: Find the matrix A relative to the basis B . T: R2 R 2 , T x , y x 2,4, y x B 2,1,1,1
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T Example 12: Let B 1, 1 , 2,1 and B 1,1 , 1, 2 be bases for R 2 , v 1 4 , and let B 2 1 A be the matrix for TRR: 2 2 relative to B . 0 1 a. Find the transition matrix P from B to B .
T b. Use the matrices P and A to find v and T v where v 1 4 . B B B
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DEFINITION OF SIMILAR MATRICES For square matrices A and A of order n , A is said to be similar to A when there exists an invertible matrix P such that APAP 1 .
THEOREM 3.30 Let A , B , and C be square matrices of order n . Then the following properties are true.
1. A is ______to ______.
2. If A is similar to B , then _____ is ______to ____.
3. If A is similar to B and B is similar to C , then _____ is ______to _____. Proof:
Example 13: Use the matrix P to show that A and A are similar. 1 0 0 2 0 0 2 0 0 A 0 1 0 A 1 1 0 P 1 1 0 , , 1 1 1 0 0 3 2 2 3
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DIAGONAL MATRICES Diagonal matrices have many ______advantages over nondiagonal matrices.
d1 0 0 __ 0 0 0d 0 0 __ 0 D 2 D k 0 0 d n 0 0 __
Also, a diagonal matrix is its own ______. Additionally, if all the diagonal elements are nonzero, then the inverse of a diagonal matrix is the matrix whose main diagonal elements are the
______of corresponding elements in the original matrix. Because of these advantages, it is important to find ways (if possible) to choose a basis for _____ such that the ______matrix is ______. 3 1 1 2 2 1 1 Example 14: Suppose A 2 is the matrix for TRR: 3 3 relative to the standard basis. 2 2 1 5 1 2 2 Find the diagonal matrix A for T relative to the basis B 1,1, 1 , 1, 1,1 , 1,1,1 .
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Example 15: Prove that if A is idempotent and B is similar to A , then B is idempotent. (An n n matrix is idempotent when AA 2 ). Proof:
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4.1: INNER PRODUCT SPACES Learning Objectives: 1. Find the length of a vector and find a unit vector 2. Find the distance between two vectors 3. Find a dot product and the angle between two vectors, determine orthogonality, and verify the Cauchy-Schwartz Inequality, the triangle inequality, and the Pythagorean Theorem 4. Use a matrix product to represent a dot product 5. Determine whether a function defines an inner product, and find the inner product of two vectors in n R , M m, n , Pn , and C a, b 6. Find an orthogonal projection of a vector onto another vector in an inner product space
DEFINITION OF LENGTH OF A VECTOR IN Rn
The ______, or ______of a vector v v1,,..., v 2 vn in ______is given by
When would the length of a vector equal to 0?
Example 1: Consider the following vectors: 1 1 u 1, v 2, 2 2 a. Find u
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b. Find v
c. Find u v
d. Find u v
e. Find 3u
f. Find 3 u
Any observations?
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THEOREM 4.1: LENGTH OF A SCALAR MULTIPLE Let v be a vector in R n and let c be a scalar. Then
where _____ is the ______of c .
Proof:
THEOREM 4.2: UNIT VECTOR IN THE DIRECTION OF v If v is a nonzero vector in R n , then the vector
has length _____ and has the same ______as v .
Proof:
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Example 2: Find the vector v with v 3 and the same direction as u 0, 2,1, 1 .
DEFINITION OF DISTANCE BETWEEN TWO VECTORS The distance between two vectors uand v in R n is
Example 3: Find the distance between u 1,1,2 and v 1,3,0 .
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DEFINITION OF THE ANGLE BETWEEN TWO VECTORS IN R n The ______between two nonzero vectors in R n is given by
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Example 4: Find the angle between u 2, 1 and v 2,0 .
DEFINITION OF DOT PRODUCT IN R n
The dot product of u u1, u 2 ,..., un and v v1, v 2 ,..., vn is the ______quantity
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Example 5: Consider the following vectors: u 1,2 v 2, 2 a. Find u v
b. Find v v
2 c. Find u
d. Find u v v
e. Find u5 v
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THEOREM 4.3: PROPERTIES OF THE DOT PRODUCT If u , v and w are vectors in R n , and c is a scalar, then the following properties are true. u v ______u v w ______c u v ______ v v ______v v 0, and v v 0 iff ______.
Example 6: Find 3u v u 3 v given that u u 8 , u v 7 , and v v 6 .
THEOREM 4.4: THE CAUCHY-SCWARZ INEQUALITY If u and v are vectors in R n , then
where ______denotes the ______value of u v .
Proof:
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Example 7: Verify the Cauch-Schwarz Inequality for u 1,0 and v 1,1 .
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DEFINITION OF ORTHOGONAL VECTORS Two vectors u and v in R n are orthogonal if
Example 7: Determine all vectors in R 2 that are orthogonal to u 3,1 .
THEOREM 4.5: THE TRIANGLE INEQUALITY If u and v are vectors in R n , then
Proof:
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THEOREM 4.6: THE PYTHAGOREAN THEOREM If u and v are vectors in R n , then u and v are orthogonal if and only if
Example 8: Verify the Pythagoren Theorem for the vectors u 3, 2 and v 4,6 .
DEFINITION OF AN INNER PRODUCT Let u , v , and w be vectors in a vector space V , and let c be any scalar. An inner product on V is a function that associates a real number u, v with each pair of vectors u and v and satisfies the following axioms. 1. u,______v
2. u,______v w
3. c u,______v
4. v, v 0, and v , v 0 iff______
NOTE:
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3 Example 8: Show that the function u, v u1 v 1 2 u 2 v 2 u 3 v 3 defines an inner product on R , where , u u1,, u 2 u 3 and v v1,, v 2 v 3 .
3 Example 9: Show that the function u, v u1 v 1 u 2 v 2 u 3 v 3 does not define an inner product on R , where , u u1,, u 2 u 3 and v v1,, v 2 v 3 .
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THEOREM 4.7: PROPERTIES OF INNER PRODUCTS Let u , v , and w be vectors in an inner product space V , and let c be any real number. 1. 0,______v
2. u v,______w Proof:
u,______c v
DEFINITION OF LENGTH, DISTANCE, AND ANGLE Let u and v be vectors in an inner product space V .
1. The length (or ______) of u is ______.
2. The distance between u and v is ______.
3. The angle between and two vectors u and v is given by
______.
4. u and v are orthogonal when ______.
If ______, then u is called a ______vector. Moreover, if v is any nonzero vector in an inner product space V , then the vector ______is a ______vector and is called the ______vector in the ______of v .
Inner product on C a, b is f, g ______.
Inner product on M 2,2 is AB,______ .
Inner product on Pn is p q ______, where
______and ______.
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Example 10: Consider the following inner product defined on R n : u 0, 6 , v 1,1 , and u, v u1 v 1 2 u 2 v 2 a. Find u, v
b. Find u
c. Find v
d. Find d u, v
Example 11: Consider the following inner product defined:
1 f, g f x g x dx , f x x , g x x2 x 2 1 a. Find f, g
b. Find f
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c. Find g
d. Find d f, g
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THEOREM 4.8 Let u and v be vectors in an inner product space V .
Cauchy-Schwarz Inequality: ______
Triangle Inequality: ______
Pythagorean Theorem: u and v are orthogonal if and only if
______
0 1 1 1 Example 12: Verify the triangle inequality for A , B , and 2 1 2 2
A, B a11 b 11 a 21 b 21 a 12 b 12 a 22 b 22 .
DEFINITION OF ORTHOGONAL PROJECTION Let u and v be vectors in an inner product space V , such that v 0 . Then the orthogonal projection of u onto v is
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THEOREM 5.9: ORTHOGONAL PROJECTION AND DISTANCE Let u and v be vectors in an inner product space V , such that v 0 . Then
Example 13: Consider the vectors u 1, 2 and v 4, 2 . Use the Euclidean inner product to find the following:
a. projvu
b. proju v
c. Sketch the graph of both projvu and proju v .
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4.2: ORTHONORMAL BASES: GRAM-SCHMIDT PROCESS Learning Objectives: 1. Show that a set of vectors is orthogonal and forms an orthonormal basis, and represent a vector relative to an orthonormal basis 2. Apply the Gram-Schmidt orthonormalization process Consider the standard basis for R3 , which is
This set is the standard basis because it has useful characteristics such as…The three vectors in the basis are
______, and they are each ______.
DEFINITIONS OF ORTHOGONAL AND ORTHONORMAL SETS A set S of a vector space V is called orthogonal when every pair of vectors in S is orthogonal. If, in addition, each vector in the set is a unit vector, then S is called
______. For S v1,,..., v 2 v n , this definition has the following form. ORTHOGONAL ORTHONORMAL
If _____ is a ______, then it is an ______basis or an ______basis, respectively.
THEOREM 4.10: ORTHOGONAL SETS ARE LINEARLY INDEPENDENT
If S v1,,..., v 2 v n is an orthogonal set of ______vectors in an inner product space V , then S is linearly independent. Proof:
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THEOREM 4.10: COROLLARY If V is an inner product space of dimension n , then any orthogonal set of n nonzero vectors is a basis for V .
Example 1: Consider the following set in R 4 . 10 3 10 3 10 10 ,0,0, ,0,0,1,0,0,1,0,0, ,0,0, 10 10 10 10 a. Determine whether the set of vectors is orthogonal.
b. If the set is orthogonal, then determine whether it is also orthonormal.
c. Determine whether the set is a basis for R n .
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THEOREM 4.11: COORDINATES RELATIVE TO AN ORTHONORMAL BASIS
If B v1,,..., v 2 v n is an orthonormal basis for an inner product space V , then the coordinate representation of a vector w relative to B is
Proof:
The coordinates of ______relative to the ______basis ____ are called the
______coefficients of ____ relative to ____. The corresponding coordinate matrix of ___ relative to ____ is
Example 2: Show that the set of vectors 2, 5 , 10, 4 in R 2 is orthogonal and normalize the set to produce an orthonormal set.
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Example 3: Find the coordinate matrix of x 3, 4 relative to the orthonormal basis 5 2 5 2 5 5 in R 2 . B ,,, 5 5 5 5
THEOREM 4.12: GRAM-SCHMIDT ORTHONORMALIZATION PROCESS
Let B v1,,..., v 2 v n be a basis for an inner product V .
Let B w1,,..., w 2 w n , where w i is given by w1 v 1
v2, w 1 w2 v 2 w 1 w1, w 1
v3,, w 1 v 3 w 2 w3 v 3 w 1 w 2 w1,, w 1 w 2 w 2
vn,,, w1 v n w 2 v n w n 1 wn v n w1 w 2 w n 1 w1,,, w 1 w 2 w 2 wn 1 w n 1
w i Let u i . Then the set B u1,,..., u 2 u n is an orthonormal basis for V . Moreover, w i spanv1 , v 2 ,..., vk span u 1 , u 2 ,..., u k for k 1,2,..., n .
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Example 4: Apply the Gram-Schmidt orthonormalization process to transform the basis B 1,0,0 , 1,1,1 , 1,1, 1 for a subspace in R 3 into an orthonormal basis. Use the Euclidean inner product on R 3 and use the vectors in the order they are given.
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4.3: MATHEMATICAL MODELS AND LEAST SQUARES ANALYSIS Learning Objectives: 1. When you are done with your homework you should be able to… 2. Define the least squares problem 3. Find the orthogonal complement of a subspace and the projection of a vector onto a subspace 4. Find the four fundamental subspaces of a matrix 5. Solve a least squares problem 6. Use least squares for mathematical modeling
In this section we will study ______systems of linear equations and learn how to find the
______of such a system.
LEAST SQUARES PROBLEM
Given an m n matrix A and a vector b in R m , the ______problem is to find ____ in R m such that ______is ______.
DEFINITION OF ORTHOGONAL SUBSPACES n The subspaces S1 and S2 of R are orthogonal when ______for all v1 in S1 and v2 in S2 .
Example 1: Are the following subspaces orthogonal? 0 1 0 S span 1 , 0 S span 1 1 and 2 1 0 1
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DEFINITION OF ORTHOGONAL COMPLEMENT If S is a subspace of R n , then the orthogonal complement of S is the set
What’s the orthogonal complement of 0 in R n ?
What’s the orthogonal complement of R n ?
DEFINITION OF DIRECT SUM n Let S1 and S2 be two subspaces of R . If each vector ______can be uniquely written as the sum of a vector ____ from ____ and a vector ___ from _____, ______, then _____ is the direct sum of ____ and ____, and you can write ______.
Example 2: Find the orthogonal complement S , and find the direct sum SS . 0 1 S span 1 1
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THEOREM 4.13: PROPERTIES OF ORTHOGONAL SUBSPACES Let S be a subspace of R n , Then the following properties are true. 1. ______
2. ______
3. ______
THEOREM 4.14: PROJECTION ONTO A SUBSPACE n n If u1,,..., u 2 ut is an orthonormal basis for the subspace S of R , and v R , then
Example 3: Find the projection of the vector v onto the subspace S . 1 0 0 1 2 0 0 1 S span , , , v 0 1 0 1 0 0 1 1
THEOREM 4.15: ORTHOGONAL PROJECTION AND DISTANCE n n Let S be a subspace of R and let v R . Then, for all u S , u projS v ,
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FUNDAMENTAL SUBSPACES OF A MATRIX Recall that if A is an m n matrix, then the column space of A is a ______of _____ consisting of all vectors of the form _____, ______. The four fundamental subspaces of the matrix A are defined as follows.
______= nullspace of A ______= nullspace of AT
______= column space of A ______= column space of AT
Example 4: Find bases for the four fundamental subspaces of the matrix 0 1 1 A 1 2 0 . 1 1 1
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THEOREM 4.16: FUNDAMENTAL SUBSPACES OF A MATRIX If A is an m n matrix, then
______and ______are orthogonal subspaces of _____.
______and ______are orthogonal subspaces of _____.
______
______
SOLVING THE LEAST SQUARES PROBLEM Recall that we are attempting to find a vector x that minimizes ______, where A is an m n matrix and b is a vector in R m . Let S be the column space of A : ______. Assume that b is not in S , because otherwise the system Ax b would be ______. We are looking for a vector ______in ____ that is as close as possible to ___. This desired vector is the ______of _____ onto ____. So, ______and ______= ______is orthogonal to ______. However, this implies that ______is in ______, which equals ______. So, ______is in the ______of _____.
The solution of the least squares problem comes down to solving the ______linear system of equations
______. These equations are called the ______equations of the least squares problem ______.
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Example 5: Find the least squares solution of the system Ax b . 1 1 1 2 1 1 1 1 A , b 0 1 1 0 1 0 1 2
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Example 6: The table shows the numbers of doctoral degrees y (in thousands) awarded in the United States from 2005 through 2008. Find the least squares regression line for the data. Then use the model to predict the number of degrees awarded in 2015. Let t represent the year, with t = 5 corresponding to 2005. (Source: U.S. National Center for Education Statistics) Year 2005 2006 2007 2008 Doctoral Degrees, y 52.6 56.1 60.6 63.7
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4.4: EIGENVALUES AND EIGENVECTORS, AND DIAGONALIZING MATRICES Learning Objectives: 1. Verify eigenvalues and corresponding eigenvectors 2. Find eigenvectors and corresponding eigenspaces 3. Use the characteristic equation to find eigenvalues and eigenvectors, and find the eigenvalues and eigenvectors of a triangular matrix 4. Find the eigenvalues and eigenvectors of a linear transformation
THE EIGENVALUE PROBLEM One of the most important problems in linear algebra is the eigenvalue problem. When A is an n n , do nonzero vectors x in R n exist such that Ax is a ______multiple of x ? The scalar, denoted by _____
(______), is called an ______of the matrix A , and the nonzero vector x is called an
______of A corresponding to .
DEFINITIONS OF EIGENVALUE AND EIGENVECTOR Let A be an n n matrix. The scalar ____ is called an ______of A when there is a
______vector x such that ______. The vector x is called an ______of A corresponding to .
*Note that an eigenvector cannot be _____. Why not?
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Example 1: Determine whether x is an eigenvector of A . 3 10 A 5 2 a. x 4, 4
b. x 8, 4
c. x 4,8
d. x 5, 3
THEOREM 4.17: EIGENVECTORS OF FORM A SUBSPACE If A is an n n matrix with an eigenvalue , then the set of all eigenvectors of , together with the zero vector is a subspace of R n . This subspace is called the ______of .
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Example 2: Find the eigenvalue(s) and corresponding eigenspace(s) of A . 1 k A 0 1
THEOREM 4.18: EIGENVALUES AND EIGENVECTORS OF A MATRIX Let A be an n n matrix. 1. An eigenvalue of A is a scalar such that ______.
2. The eigenvectors of A corresponding to are the ______solutions of
______.
* The equation ______is called the ______of
A . When expanded to polynomial form, the polynomial is called the ______
______of A . This definition tells you that the ______of an n n matrix
A correspond to the ______of the characteristic polynomial of A .
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Example 3: Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectos) of the matrix. 3 2 1 A 0 0 2 0 2 0
THEOREM 4.19: EIGENVALUES OF TRIANGULAR MATRICES If A is an n n triangular matrix, then its eigenvalues are the entries on its main ______.
Example 4: Find the eigenvalues of the triangular matrix. 5 0 0 3 7 0 4 2 3
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EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS
A number is called an ______of a linear transformation ______when there is a
______vector ____ such that ______. The vector x is called an ______of T corresponding to , and the set of all eigenvectors of (with the zero vector) is called the
______of .
Example 6: Consider the linear transformation TRR: n n whose matrix A relative to the standard base is given. Find (a) the eigenvalues of A , (b) a basis for each of the corresponding eigenspaces, and (c) the matrix A for T relative to the basis B , where B is made up of the basis vectors found in part b). 6 2 A 3 1
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4.5: DIAGONALIZATION Learning Objectives: 1. Find the eigenvectors of similar matrices, determine whether a matrix A is diagonalizable, and find a matrix P such that PAP1 is diagonal 2. Find, for a linear transformation TVV: , a basis B for V such that the matrix for T relative to B is diagonal
DEFINITION OF A DIAGONALIZABLE MATRIX An n n matrix A is diagonalizable when A is similar to a diagonal matrix. That is, A is diagonalizable when there exists an invertible matrix ____ such that ______is a diagonal matrix.
THEOREM 4.20: SIMILAR MATRICES HAVE THE SAME EIGENVALUES If A and B are similar n n matrices, then the have the same ______.
Proof:
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Example 1: (a) verify that A is diagonalizable by computing PAP1 , and (b) use the result of part (a) and Theorem 4.20 to find the eigenvalues of A . 1 3 3 1 AP , 1 5 1 1
THEOREM 4.21: CONDITION FOR DIAGONALIZATION An n n matrix A is diagonalizable if and only if it has n ______eigenvectors.
Proof:
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Example 2: For the matrix A , find, if possible, a nonsingular matrix P such that PAP1 is diagonal. Verify PAP1 is a diagonal matrix with the eigenvalues on the main diagonal. 4 0 0 A 2 2 0 0 2 2
STEPS FOR DIAGONALIZING AN n n SQUARE MATRIX Let A be an n n matrix. 1. Find n linearly independent eigenvectors ______for A (if possible) with
corresponding eigenvalues ______. If n linearly independent eigenvectors do not
exist, then A is not diagonalizable.
2. Let P be the n n matrix whose columns consist of these eigenvectors. That is, ______. The diagonal matrix ______will have the eigenvalues
______on its main ______(and ______elsewhere). Note that
the order of the eigenvectors used to form P will determine the order in which the eigenvalues
appear on the main ______of _____.
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THEOREM 4.22: SUFFICIENT CONDITION FOR DIAGONALIZATION If an n n matrix A has ______eigenvalues, then the corresponding eigenvectors are
______and A is ______.
Proof:
Example 3: Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. 2 0 5 2
Example 4: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. TR:3 RTxyz 3 :,, 223,2 x yzxyzxy 6, 2
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4.5: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION Learning Objectives: 1. Recognize, and apply properties of, symmetric matrices 2. Recognize, and apply properties of, orthogonal matrices 3. Find an orthogonal matrix P that orthogonally diagonalizes a symmetric matrix A
SYMMETRIC MATRICES Symmetric matrices arise more often in ______than any other major class of matrices.
The theory depends on both ______and ______. For most matrices, you need to go through most of the diagonalization ______to ascertain whether a matrix is ______. We learned about one exception, a ______matrix, which has ______entries on the main ______. Another type of matrix which is guaranteed to be ______is a ______matrix.
DEFINITION OF SYMMETRIC MATRIX
A square matrix A is ______when it is equal to its ______:______.
Example 1: Determine which of the matrices below are symmetric. 6 5 4 1 2 3 4 2 5 3 2 1 2 7 1 0 B 5 1 0 A , , C , D 5 1 1 2 3 3 1 7 2 4 0 1 4 0 2 5
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Example 2: Using the diagonalization process, determine if A is diagonalizable. If so, diagonalize the matrix A . 6 1 A 1 5
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THEOREM 4.23: PROPERTIES OF SYMMETRIC MATRICES If A is an n n symmetric matrix, then the following properties are true. 1. A is ______.
2. All ______of A are ______.
3. If is an ______of A with multiplicity ____, then
____ has ____ linearly ______eigenvectors. That is, the
______of has dimension ____. Proof of Property 1 (for a 2 x 2 symmetric matrix):
Example 3: Prove that the symmetric matrix is diagonalizable. a a a A a a a a a a
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Example 4: Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. 2 1 1 A 1 2 1 1 1 2
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DEFINITION OF AN ORTHOGONAL MATRIX A square matrix P is ______when it is ______and when
______.
THEOREM 4.24: PROPERTY OF ORTHOGONAL MATRICES
An n n matrix P is orthogonal if and only if its ______vectors form an
______set.
Example 5: Determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. 4 3 0 5 5 A 0 1 0 3 4 0 5 5
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THEOREM 4.25: PROPERTY OF SYMMETRIC MATRICES
Let A be an n n symmetric matrix. If 1 and 2 are ______eigenvalues of A , then their corresponding ______x1 and x2 are ______.
THEOREM 4.26: FUNDAMENTAL THEOREM OF SYMMETRIC MATRICES Let A be an n n matrix. Then A is ______and has ______eigenvalues if and only if A is ______.
Proof:
STEPS FOR DIAGONALIZING A SYMMETRIC MATRIX Let A be an n n symmetric matrix. 1. Find all ______of A and determine the ______of each.
2. For ______eigenvalue of multiplicity _____, find a ______eigenvector. That is, find any
______and then ______it.
3. For ______eigenvalue of multiplicity ______, find a set of ______
______eigenvectors. If this set is not ______, apply the
______process.
4. The results of steps 2 and 3 produce an ______set of ______eigenvectors. Use
these eigenvectors to form the ______of ______. The matrix ______
will be ______. The main entries of _____ are the ______of _____.
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Example 5: Find a matrix P such that PT AP orthogonally diagonalizes A . Verify that PT AP gives the proper diagonal form. 0 1 1 A 1 0 1 1 0 0
Example 6: Prove that if a symmetric matrix A has only one eigenvalue , then AI .
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4.6: APPLICATIONS OF EIGENVALUES AND EIGENVECTORS Learning Objectives: 1. Find the matrix of a quadratic form and use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric
QUADRATIC FORMS Every conic section in the xy-plane can be written as:
If the equation of the conic has no xy-term (______), then the axes of the graphs are parallel to the coordinate axes. For second-degree equations that have an xy-term, it is helpful to first perform a
a c ______of axes that eliminates the xy-term. The required rotation angle is cot 2 . With b
2 this rotation, the standard basis for R , ______is rotated to form the new basis
______.
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Example 1: Find the coordinates of a point x, y in R2 relative to the basis B cos ,sin , sin ,cos .
ROTATION OF AXES The general second-degree equation ax2 bxy cy 2 dx ey f 0 can be written in the form 2 2 a x c y d x e y f 0 by rotating the coordinate axes counterclockwise through the angle a c , where is defined by cot 2 . The coefficients of the new equation are obtained from the b substitutions x xcos y sin and y xsin y cos .
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Example 2: Perform a rotation of axes to eliminate the xy-terms in 5x2 6 xy 5 y 2 142 x 22 y 180 . Sketch the graph of the resulting equation.
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______and ______can be used to solve the rotation of axes problem. It turns out that the coefficients a’ and c’ are eigenvalues of the matrix
The expression ______is called the ______form associated with the quadratic equation and the matrix ____ is called the ______of the ______form. Note that ____ is ______. Moreover, ___ will be ______if and only if its corresponding quadratic form has no ______term.
Example 3: Find the matrix of quadratic form associated with each quadratic equation. a. x24 y 2 4 0
2 2 b. 5x 6 xy 5 y 142 x 22 y 180
Now, let’s check out how to use the matrix of quadratic form to perform a rotation of axes. x 2 2 Let X . Then the quadratic expression ax bxy cy dx ey f can be written in matrix form as y follows:
If ______, then no ______is necessary. But if ______, then because ___ is symmetric, you may conclude that there exists an ______matrix ____ such that ______is diagonal. So, if you let
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then it follows that ______, and
The choice of ___ must be made with care. Since ___ is orthogonal, its determinant will be ______. If P is chosen so that P 1, then P will be of the form
where gives the angle of rotation of the conic measured from the ______x-axis to the positive x’-axis.
PRINCIPAL AXES THEOREM 2 2 For a conic whose equation is ax bxy cy dx ey f 0 , the rotation given by ______eliminates the xy-term when P is an orthogonal matrix, with P 1, that diagonalizes A . That is
where 1 and 2 are eigenvalues of A . The equation of the rotated conic is given by
Example 4: Use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 5x2 6 xy 5 y 2 142 x 22 y 180
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