DOCSLIB.ORG

8.3 Vector Spaces and Subspaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

8.3 Vector Spaces and Subspaces Performance Criterion: 8. (d) Determine whether a subset of Rn is a subspace. If so, prove it; if not, give an appropriate counterexample. Vector Spaces The term “space” in math simply means a set of objects with some additional special properties. There are metric spaces, function space, topological spaces, Banach spaces, and more. The vectors that we have been dealing with make up the vector spaces called R2, R3 and, for larger values, Rn. In general, a vector space is simply a collection of objects called vectors (and a set of scalars) that satisfy certain properties. Definition 8.3.1: Vector Space A vector space is a set V of objects called vectors and a set of scalars (usually the real numbers R), with the operations of vector addition and scalar multiplication, for which the following properties hold for all u, v, w in V and scalars c and d. 1. u + v is in V 2. u + v = v + u 3. (u + v)+ w = u + (v + w) 4. There exists a vector 0 in V such that u + 0 = u. This vector is called the zero vector. 5. For every u in V there exists a vector −u in V such that u + (−u) = 0. 6. cu is in V . 7. c(u + v)= cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1u = u Note that items 1 and 6 of the above definition say that the vector space V is closed under addition and scalar multiplication. When working with vector spaces, we will be very interested in certain subsets of those vector spaces that are the span of a set of vectors. As you proceed, recall Example 8.2(b), where we showed that the span of a set of vectors is closed under addition and scalar multiplication. Subspaces of Vector Spaces As you should know by now, the two main operations with vectors are multiplication by scalars and addition of vectors. (Note that these two combined give us linear combinations, the foundation of almost everything we’ve done.) A given vector space can have all sorts of subsets; consider the following subsets of R2. • The set S1 consisting of the first quadrant and the nonnegative parts of the two axes, or all vectors of the x1 form such that x1 ≥ 0 and x2 ≥ 0. x2 110 3 • The set S2 consisting of the line containing the vector . Algebraically this is all vectors of the form 2 3 t where t ranges over all real numbers. 2 • The set S3 consisting of the first and third quadrants and both axes. This can be described as the set of all x1 vectors with x1x2 ≥ 0. x2 Our current concern is whether these subsets of R2 are closed under addition and scalar multiplication. With a bit of thought you should see that S1 is closed under addition, but not scalar multiplication when the scalar is negative: S1 S1 u + v u w v cw c< 0 In some sense we can solve the problem of not being closed under scalar multiplication by including the third quadrant as well to get S3, but then the set isn’t closed under addition: S3 S3 u w S3 cw u + v S3 c< 0 v Finally, the set S2 is closed under both addition and scalar multiplication. That is a bit messy to show with a 3 diagram, but consider the following. S2 is the span of the single vector , and we showed in the last section 2 that the span of any set of vectors is closed under addition and scalar multiplication. It turns out that when working with vector spaces the only subsets of any real interest are the ones that are closed under both addition and scalar multiplication. We give such subsets a name: Definition 8.3.2: Subspace of Rn A subset S of Rn is called a subspace of Rn if for every scalar c and any vectors u and v in S, cu and u + v are also in S. That is, S is closed under scalar multiplication and addition. You will be asked whether certain subsets of R2, R3 or Rn are subspaces, and it is your job to back your answers up with some reasoning. This is done as follows: • When a subset IS a subspace a general proof is required. That is, we must show that the set is closed under scalar multiplication and addition, for ALL scalars and ALL vectors. We may have to do this outright, but if it is clear that the set of vectors is the span of some set of vectors, then we know from the argument presented in Example 8.2(b) that the set is closed under addition and scalar multiplication, so it is a subspace. • When a subset IS NOT a subspace, we demonstrate that fact with a SPECIFIC example. Such an example is called a counterexample. Notice that all we need to do to show that a subset is not a subspace is to show 111 that that it is not closed under scalar multiplication OR vector addition. If either is the case, then the set in question is not a subspace. Even if both are the case, we need only show one. The following examples illustrate these things. x1 ⋄ Example 8.3(a): Show that the set S1 consisting of all vectors of the form such that x1 ≥ 0 and x2 2 x2 ≥ 0 is not a subspace of R . As mentioned before, this set is not closed under multiplication by negative scalars, so we just need to give a specific 3 3 −6 example of this. Let u = and c = −2. Clearly u is in S1 and cu = (−2) = , which is 5 5 −10 2 not in S1. Therefore S1 is not closed under scalar multiplication so it is not a subspace of R . ♠ 3 ⋄ Example 8.3(b): Show that the set S2 consisting of all vectors of the form t , where t ranges over 2 all real numbers, is a subspace of R2. 3 3 Let c be any scalar and let u = s and v = t . Then u and v are both in S2 and 2 2 3 3 3 3 3 cu = c s = (cs) and u + v = s + t = (s + t) . 2 2 2 2 2 2 Because cs and s + t are scalars, we can see that both cu and u + v are in S2, so S2 is a subspace of R . ♠ This last example demonstrates the general method for showing that a set of vectors is closed under addition and scalar multiplication. That said, the given subspace could have been shown to be s subspace by simply observing 3 that it is the span of the set consisting of the single vector . From Theorem 8.2.2 we know that the span of 2 any set of vectors is a subspace, so the set described in the above example is a subspace of R2. ⋄ Example 8.3(c): Determine whether the subset S of R3 consisting of all vectors of the form x = 2 4 5 + t −1 is a subspace. If it is, prove it. If it is not, provide a counterexample. −1 3 We recognize this as a line in R3 passing through the point (2, 5, −1), and it is not hard to show that the line does not pass through the origin. Remember l that what we mean by the line is really all position vectors (so with tails at the origin) whose tips are on the line. Considering a similar situation in R2, we see that u is such a vector for the line l shown. It should be clear that if we u multiply u by any scalar other than one, the resulting vector’s tip will not lie on the line. Thus we would guess that the set S, even though it is in R3, is probably not closed under scalar multiplication. 2 4 6 Now let’s prove that it isn’t. To do this we first let u = 5 +1 −1 = 4 , which is in S. Let −1 3 2 12 c =2, so 2u = 8 . We need to show that this vector is not in S. If it were, there would have to be a scalar 4 12 2 4 2 10 4 t such that 8 = 5 + t −1 . Subtracting 5 we get 3 = t −1 . We can see 4 −1 3 −1 5 3 that the value of tthat would be needed to give the correct second component would be −3, but this would 112 12 result in a third component of −9, which is not correct. Thus there is no such t and the vector 8 is not 4 3 in S. Thus S is not closed under scalar multiplication, so it is not a subspace of R . ♠ We should compare the results of Examples 8.3(b) and 8.3(c). Note that both are lies in their respective Rn’s, but the line in 8.3(b) passes through the origin, and the one in 8.3(c) does not. It is no coincidence that the set in 8.3(b) is a subspace and the set in 8.3(c) is not. If a set S is a subspace, being closed under scalar multiplication means that zero times any vector in the subspace must also be in the subspace. But zero times a vector is the zero vector 0.
Recommended publications
  • Db Math Marco Zennaro Ermanno Pietrosemoli Goals

    Db Math Marco Zennaro Ermanno Pietrosemoli Goals

    dB Math Marco Zennaro Ermanno Pietrosemoli Goals ‣ Electromagnetic waves carry power measured in milliwatts. ‣ Decibels (dB) use a relative logarithmic relationship to reduce multiplication to simple addition. ‣ You can simplify common radio calculations by using dBm instead of mW, and dB to represent variations of power. ‣ It is simpler to solve radio calculations in your head by using dB. 2 Power ‣ Any electromagnetic wave carries energy - we can feel that when we enjoy (or suffer from) the warmth of the sun. The amount of energy received in a certain amount of time is called power. ‣ The electric field is measured in V/m (volts per meter), the power contained within it is proportional to the square of the electric field: 2 P ~ E ‣ The unit of power is the watt (W). For wireless work, the milliwatt (mW) is usually a more convenient unit. 3 Gain and Loss ‣ If the amplitude of an electromagnetic wave increases, its power increases. This increase in power is called a gain. ‣ If the amplitude decreases, its power decreases. This decrease in power is called a loss. ‣ When designing communication links, you try to maximize the gains while minimizing any losses. 4 Intro to dB ‣ Decibels are a relative measurement unit unlike the absolute measurement of milliwatts. ‣ The decibel (dB) is 10 times the decimal logarithm of the ratio between two values of a variable. The calculation of decibels uses a logarithm to allow very large or very small relations to be represented with a conveniently small number. ‣ On the logarithmic scale, the reference cannot be zero because the log of zero does not exist! 5 Why do we use dB? ‣ Power does not fade in a linear manner, but inversely as the square of the distance.
  • The Five Fundamental Operations of Mathematics: Addition, Subtraction

    The Five Fundamental Operations of Mathematics: Addition, Subtraction

    The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more.
  • Eigenvalues and Eigenvectors

    Eigenvalues and Eigenvectors

    Jim Lambers MAT 605 Fall Semester 2015-16 Lecture 14 and 15 Notes These notes correspond to Sections 4.4 and 4.5 in the text. Eigenvalues and Eigenvectors In order to compute the matrix exponential eAt for a given matrix A, it is helpful to know the eigenvalues and eigenvectors of A. Definitions and Properties Let A be an n × n matrix. A nonzero vector x is called an eigenvector of A if there exists a scalar λ such that Ax = λx: The scalar λ is called an eigenvalue of A, and we say that x is an eigenvector of A corresponding to λ. We see that an eigenvector of A is a vector for which matrix-vector multiplication with A is equivalent to scalar multiplication by λ. Because x is nonzero, it follows that if x is an eigenvector of A, then the matrix A − λI is singular, where λ is the corresponding eigenvalue. Therefore, λ satisfies the equation det(A − λI) = 0: The expression det(A−λI) is a polynomial of degree n in λ, and therefore is called the characteristic polynomial of A (eigenvalues are sometimes called characteristic values). It follows from the fact that the eigenvalues of A are the roots of the characteristic polynomial that A has n eigenvalues, which can repeat, and can also be complex, even if A is real. However, if A is real, any complex eigenvalues must occur in complex-conjugate pairs. The set of eigenvalues of A is called the spectrum of A, and denoted by λ(A). This terminology explains why the magnitude of the largest eigenvalues is called the spectral radius of A.
  • Algebra of Linear Transformations and Matrices Math 130 Linear Algebra

    Algebra of Linear Transformations and Matrices Math 130 Linear Algebra

    Then the two compositions are 0 −1 1 0 0 1 BA = = 1 0 0 −1 1 0 Algebra of linear transformations and 1 0 0 −1 0 −1 AB = = matrices 0 −1 1 0 −1 0 Math 130 Linear Algebra D Joyce, Fall 2013 The products aren't the same. You can perform these on physical objects. Take We've looked at the operations of addition and a book. First rotate it 90◦ then flip it over. Start scalar multiplication on linear transformations and again but flip first then rotate 90◦. The book ends used them to define addition and scalar multipli- up in different orientations. cation on matrices. For a given basis β on V and another basis γ on W , we have an isomorphism Matrix multiplication is associative. Al- γ ' φβ : Hom(V; W ) ! Mm×n of vector spaces which though it's not commutative, it is associative. assigns to a linear transformation T : V ! W its That's because it corresponds to composition of γ standard matrix [T ]β. functions, and that's associative. Given any three We also have matrix multiplication which corre- functions f, g, and h, we'll show (f ◦ g) ◦ h = sponds to composition of linear transformations. If f ◦ (g ◦ h) by showing the two sides have the same A is the standard matrix for a transformation S, values for all x. and B is the standard matrix for a transformation T , then we defined multiplication of matrices so ((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x)) = f(g(h(x))) that the product AB is be the standard matrix for S ◦ T .
  • Operations with Algebraic Expressions: Addition and Subtraction of Monomials

    Operations with Algebraic Expressions: Addition and Subtraction of Monomials

    Operations with Algebraic Expressions: Addition and Subtraction of Monomials A monomial is an algebraic expression that consists of one term. Two or more monomials can be added or subtracted only if they are LIKE TERMS. Like terms are terms that have exactly the SAME variables and exponents on those variables. The coefficients on like terms may be different. Example: 7x2y5 and -2x2y5 These are like terms since both terms have the same variables and the same exponents on those variables. 7x2y5 and -2x3y5 These are NOT like terms since the exponents on x are different. Note: the order that the variables are written in does NOT matter. The different variables and the coefficient in a term are multiplied together and the order of multiplication does NOT matter (For example, 2 x 3 gives the same product as 3 x 2). Example: 8a3bc5 is the same term as 8c5a3b. To prove this, evaluate both terms when a = 2, b = 3 and c = 1. 8a3bc5 = 8(2)3(3)(1)5 = 8(8)(3)(1) = 192 8c5a3b = 8(1)5(2)3(3) = 8(1)(8)(3) = 192 As shown, both terms are equal to 192. To add two or more monomials that are like terms, add the coefficients; keep the variables and exponents on the variables the same. To subtract two or more monomials that are like terms, subtract the coefficients; keep the variables and exponents on the variables the same. Addition and Subtraction of Monomials Example 1: Add 9xy2 and −8xy2 9xy2 + (−8xy2) = [9 + (−8)] xy2 Add the coefficients. Keep the variables and exponents = 1xy2 on the variables the same.
  • Math 217: Multilinearity of Determinants Professor Karen Smith (C)2015 UM Math Dept Licensed Under a Creative Commons By-NC-SA 4.0 International License

    Math 217: Multilinearity of Determinants Professor Karen Smith (C)2015 UM Math Dept Licensed Under a Creative Commons By-NC-SA 4.0 International License

    Math 217: Multilinearity of Determinants Professor Karen Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. A. Let V −!T V be a linear transformation where V has dimension n. 1. What is meant by the determinant of T ? Why is this well-defined? Solution note: The determinant of T is the determinant of the B-matrix of T , for any basis B of V . Since all B-matrices of T are similar, and similar matrices have the same determinant, this is well-defined—it doesn't depend on which basis we pick. 2. Define the rank of T . Solution note: The rank of T is the dimension of the image. 3. Explain why T is an isomorphism if and only if det T is not zero. Solution note: T is an isomorphism if and only if [T ]B is invertible (for any choice of basis B), which happens if and only if det T 6= 0. 3 4. Now let V = R and let T be rotation around the axis L (a line through the origin) by an 21 0 0 3 3 angle θ. Find a basis for R in which the matrix of ρ is 40 cosθ −sinθ5 : Use this to 0 sinθ cosθ compute the determinant of T . Is T othogonal? Solution note: Let v be any vector spanning L and let u1; u2 be an orthonormal basis ? for V = L . Rotation fixes ~v, which means the B-matrix in the basis (v; u1; u2) has 213 first column 405.
  • Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008

    Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008

    E. Hitzer, Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008. Basic Multivector Calculus Eckhard Hitzer, Department of Applied Physics, Faculty of Engineering, University of Fukui Abstract: We begin with introducing the generalization of real, complex, and quaternion numbers to hypercomplex numbers, also known as Clifford numbers, or multivectors of geometric algebra. Multivectors encode everything from vectors, rotations, scaling transformations, improper transformations (reflections, inversions), geometric objects (like lines and spheres), spinors, and tensors, and the like. Multivector calculus allows to define functions mapping multivectors to multivectors, differentiation, integration, function norms, multivector Fourier transformations and wavelet transformations, filtering, windowing, etc. We give a basic introduction into this general mathematical language, which has fascinating applications in physics, engineering, and computer science. more didactic approach for newcomers to geometric algebra. 1. Introduction G(I) is the full geometric algebra over all vectors in the “Now faith is being sure of what we hope for and certain of what we do not see. This is what the ancients were n-dimensional unit pseudoscalar I e1 e2 ... en . commended for. By faith we understand that the universe was formed at God’s command, so that what is seen was not made 1 A G (I) is the n-dimensional vector sub-space of out of what was visible.” [7] n The German 19th century mathematician H. Grassmann had grade-1 elements in G(I) spanned by e1,e2 ,...,en . For the clear vision, that his “extension theory (now developed to geometric calculus) … forms the keystone of the entire 1 vectors a,b,c An G (I) and scalars , ,, ; structure of mathematics.”[6] The algebraic “grammar” of this universal form of calculus is geometric algebra (or Clifford G(I) has the fundamental properties of algebra).
  • MATH 304 Linear Algebra Lecture 24: Scalar Product. Vectors: Geometric Approach

    MATH 304 Linear Algebra Lecture 24: Scalar Product. Vectors: Geometric Approach

    MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B′ A′ A vector is represented by a directed segment. • Directed segment is drawn as an arrow. • Different arrows represent the same vector if • they are of the same length and direction. Vectors: geometric approach v B A v B′ A′ −→AB denotes the vector represented by the arrow with tip at B and tail at A. −→AA is called the zero vector and denoted 0. Vectors: geometric approach v B − A v B′ A′ If v = −→AB then −→BA is called the negative vector of v and denoted v. − Vector addition Given vectors a and b, their sum a + b is defined by the rule −→AB + −→BC = −→AC. That is, choose points A, B, C so that −→AB = a and −→BC = b. Then a + b = −→AC. B b a C a b + B′ b A a C ′ a + b A′ The difference of the two vectors is defined as a b = a + ( b). − − b a b − a Properties of vector addition: a + b = b + a (commutative law) (a + b) + c = a + (b + c) (associative law) a + 0 = 0 + a = a a + ( a) = ( a) + a = 0 − − Let −→AB = a. Then a + 0 = −→AB + −→BB = −→AB = a, a + ( a) = −→AB + −→BA = −→AA = 0. − Let −→AB = a, −→BC = b, and −→CD = c. Then (a + b) + c = (−→AB + −→BC) + −→CD = −→AC + −→CD = −→AD, a + (b + c) = −→AB + (−→BC + −→CD) = −→AB + −→BD = −→AD. Parallelogram rule Let −→AB = a, −→BC = b, −−→AB′ = b, and −−→B′C ′ = a. Then a + b = −→AC, b + a = −−→AC ′.
  • Single Digit Addition for Kindergarten

    Single Digit Addition for Kindergarten

    Single Digit Addition for Kindergarten Print out these worksheets to give your kindergarten students some quick one-digit addition practice! Table of Contents Sports Math Animal Picture Addition Adding Up To 10 Addition: Ocean Math Fish Addition Addition: Fruit Math Adding With a Number Line Addition and Subtraction for Kids The Froggie Math Game Pirate Math Addition: Circus Math Animal Addition Practice Color & Add Insect Addition One Digit Fairy Addition Easy Addition Very Nutty! Ice Cream Math Sports Math How many of each picture do you see? Add them up and write the number in the box! 5 3 + = 5 5 + = 6 3 + = Animal Addition Add together the animals that are in each box and write your answer in the box to the right. 2+2= + 2+3= + 2+1= + 2+4= + Copyright © 2014 Education.com LLC All Rights Reserved More worksheets at www.education.com/worksheets Adding Balloons : Up to 10! Solve the addition problems below! 1. 4 2. 6 + 2 + 1 3. 5 4. 3 + 2 + 3 5. 4 6. 5 + 0 + 4 7. 6 8. 7 + 3 + 3 More worksheets at www.education.com/worksheets Copyright © 2012-20132011-2012 by Education.com Ocean Math How many of each picture do you see? Add them up and write the number in the box! 3 2 + = 1 3 + = 3 3 + = This is your bleed line. What pretty FISh! How many pictures do you see? Add them up. + = + = + = + = + = Copyright © 2012-20132010-2011 by Education.com More worksheets at www.education.com/worksheets Fruit Math How many of each picture do you see? Add them up and write the number in the box! 10 2 + = 8 3 + = 6 7 + = Number Line Use the number line to find the answer to each problem.
  • Inner Product Spaces

    Inner Product Spaces

    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
  • Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007)

    Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007)

    MAT067 University of California, Davis Winter 2007 Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007) The abstract definition of vector spaces only takes into account algebraic properties for the addition and scalar multiplication of vectors. For vectors in Rn, for example, we also have geometric intuition which involves the length of vectors or angles between vectors. In this section we discuss inner product spaces, which are vector spaces with an inner product defined on them, which allow us to introduce the notion of length (or norm) of vectors and concepts such as orthogonality. 1 Inner product In this section V is a finite-dimensional, nonzero vector space over F. Definition 1. An inner product on V is a map ·, · : V × V → F (u, v) →u, v with the following properties: 1. Linearity in first slot: u + v, w = u, w + v, w for all u, v, w ∈ V and au, v = au, v; 2. Positivity: v, v≥0 for all v ∈ V ; 3. Positive definiteness: v, v =0ifandonlyifv =0; 4. Conjugate symmetry: u, v = v, u for all u, v ∈ V . Remark 1. Recall that every real number x ∈ R equals its complex conjugate. Hence for real vector spaces the condition about conjugate symmetry becomes symmetry. Definition 2. An inner product space is a vector space over F together with an inner product ·, ·. Copyright c 2007 by the authors. These lecture notes may be reproduced in their entirety for non- commercial purposes. 2NORMS 2 Example 1. V = Fn n u =(u1,...,un),v =(v1,...,vn) ∈ F Then n u, v = uivi.
  • Matrix Multiplication. Diagonal Matrices. Inverse Matrix. Matrices

    Matrix Multiplication. Diagonal Matrices. Inverse Matrix. Matrices

    MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a11 a12 ... a1n a21 a22 ... a2n . .. . am1 am2 ... amn Notation: A = (aij )1≤i≤n, 1≤j≤m or simply A = (aij ) if the dimensions are known. Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. Negative: −A is defined as (−1)A. Subtraction: A − B is defined as A + (−B). As far as the linear operations are concerned, the m×n matrices can be regarded as mn-dimensional vectors. Properties of linear operations (A + B) + C = A + (B + C) A + B = B + A A + O = O + A = A A + (−A) = (−A) + A = O r(sA) = (rs)A r(A + B) = rA + rB (r + s)A = rA + sA 1A = A 0A = O Dot product Definition. The dot product of n-dimensional vectors x = (x1, x2,..., xn) and y = (y1, y2,..., yn) is a scalar n x · y = x1y1 + x2y2 + ··· + xnyn = xk yk . Xk=1 The dot product is also called the scalar product. Matrix multiplication The product of matrices A and B is defined if the number of columns in A matches the number of rows in B. Definition. Let A = (aik ) be an m×n matrix and B = (bkj ) be an n×p matrix.