Linear independence, Linear transformation Math 112, Week 3 Goals: • Solutions of linear system in parametric vector form. • Linear independence and linear dependence. • Introduction to Linear transformation. Suggested Textbook Readings: Sections x1.5, x1.7, x1.8, x1.9 Steps of writing the solution set (of a consistent system) in parametric vector form: 1. Row reduce the augmented matrix to RREF. 2. Express each basic variable in terms of any free variables appearing in an equation. 3. Write a typical solution ~x as a vector whose entries depend on the free variables, if any. 4. Decompose ~x into a linear combination of vectors (with numeric entries) using the free variables as parameters. Facts of linearly dependent set of vectors: 1. A set S = f~v1; ··· ;~vkg of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. 2. If a set contains more vectors than there are entries in each vector, then the set n is linearly dependent. That is, any set f~v1; ··· ;~vkg in R is linearly dependent if k > n. n 3. If a set S = f~v1; ··· ;~vkg in R contains the zero vector, then the set is linearly dependent. Week 3: Linear independence, Linear transformation 2 Example 1: Solve the system and write the solutions in vector form. 8 3x + 5x − 4x = 0 <> 1 2 3 −3x1 − 2x2 + 4x3 = 0 > : 6x1 + x2 − 8x3 = 0 Example 2: Solve the system and write the solutions in vector form. 2x1 − 3x2 − 4x3 = 0 MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 3 Example 3: Solve the system and write the solutions in vector form. 8 3x + 5x − 4x = 3 <> 1 2 3 −3x1 − 2x2 + 4x3 = 0 > : 6x1 + x2 − 8x3 = −3 Example 4: Solve the system and write the solutions in vector form. 2x1 − 3x2 − 4x3 = 1 MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 4 Example 5: Determine if the following sets of vectors are linearly indepen- dent. 2 3 2 3 2 3 2 3 5 2 1 −1 1. f4 5 ; 4 5 ; 4 5 ; 4 5g 1 8 3 4 2 3 2 3 4 6 6 7 6 7 6 7 6 7 2. f6−27 ; 6−37g 4 5 4 5 6 9 2 3 2 3 2 3 3 0 −6 6 7 6 7 6 7 6 7 6 7 6 7 3. f6 5 7 ; 607 ; 6 5 7g 4 5 4 5 4 5 −1 0 4 2 3 2 3 −8 2 6 7 6 7 6 7 6 7 4. f6 12 7 ; 6−37g 4 5 4 5 −4 −1 2 3 2 3 2 3 1 2 3 6 7 6 7 6 7 6 7 6 7 6 7 5. f627 ; 637 ; 657g 4 5 4 5 4 5 3 4 7 MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 5 Matrix transformation and Linear Transformation Definition: Let A be an m × n matrix. The transformation T : Rn ! Rm given by T (~x) = A~x is called a matrix transformation. 2 3 1 1 For example, T (~x) = A~x, where A = 4 5, is a matrix transformation from R2 to 1 −1 2 3 1 2 −1 R2, and the matrix B = 4 5 defines a transformation T : R3 ! R2, T (~x) = B~x 1 0 3 for ~x 2 R3. Definition: A transformation for which T (~u + ~v) = T (~u) + T (~v); and T (k~u) = kT (~u) where k is a scalar, is called a linear transformation. 2 3 2 3 x1 2x1 For example, T : R2 ! R2, with T (4 5) = 4 5 is a linear transformation, but x2 x2 2 3 2 3 2 x1 x1 + x2 T (4 5) = 4 5 is not a linear transformation. x2 x2 Theorem:Every matrix transformation is a linear transformation. Theorem:Every linear transformation T : Rn ! Rm has a unique matrix A so that T (~x) = A~x. This matrix is called the standard matrix of T . Standard matrix of a linear transformation: The standard matrix of a linear n m transformation T : R ! R can be found by computing T (~ej), j = 1; 2; ··· ; n, where n ~ej denotes the vector in R whose j-th entry is 1 and the rest entries are 0. The standard matrix of T is given by: h i T (~e1) T (~e2) ··· T (~en) MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 6 2 3 1 −3 6 7 6 7 Example 6: Let A = 6 3 5 7, and define a transformation 4 5 −1 7 T : R2 ! R3 by T (~x) = A~x. 2 3 2 1. Let ~u = 4 5, find T (~u), the image of ~u under T . −1 2 3 3 6 7 ~ 6 7 2 ~ 2. Let b = 6 2 7, find an ~x in R whose image under T is b. 4 5 −5 3. Is there more than one ~x whose image under T is ~b? MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 7 2 3 1 −3 6 7 6 7 A = 6 3 5 7, T (~x) = A~x 4 5 −1 7 2 3 3 6 7 6 7 4. Let ~c = 627, determine if ~c is in the range of T . 4 5 5 MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 8 Example 7: (Rotation Transformation) Let T : R2 ! R2 be the transfor- mation that rotates each point in R2 about the origin through an angle θ, with counterclockwise rotation for a positive angle. This is a linear transformation. Find its standard matrix. MATH 112 Winter 2016.