[Solved] MATH141: Linear Analysis I Homework 07

Verify that functions defined by a matrix is always linear. More precisely, verify that L_A ^: R² R², L_A(~x) = A~x, with, is linear.

1. Determine whether each of the following functions is linear or not. Explain your reasoning.
2. Assume that T : R² R² is a linear transformation. Let and. Draw the image of the

half-shaded unit square (shown below) under the given transformation T , and find the matrix A such that T = L_A.

• T stretches by a factor of 2 in the x-direction and by a factor of 3 in the y-direction.
• T is a reflection across the line y = x.
• T is a rotation (about the origin) through /4
• T is a vertical shear that maps ~e₁ into ~e₁~e₂ but leaves the vector ~e₂

4. For any given mn matrix A, we are going to use the notation L_A to denote the linear transformation that A defines, i.e., L_A : RⁿR^m : L_A(~x) = A~x. For each given matrix, answer the following questions.
   • Rewrite L_D : RⁿR^m with correct numbers for m and n filled in for each matrix. Repeat for L_E and L_F.
   • Find some way to explain in words and/or graphically what this transformation does in taking vectors from Rⁿ to R^m. You might find it helpful to try out a few input vectors and see what their image is under the transformation.
   • Is this transformation one-to-one? (Hint: Review problem#6 of Homework 06.)
     1. If so, explain which properties of the matrix make the transformation one-to-one.

MATH 141: Linear Analysis I Homework 07 Fall 2019

1. If not, given an example of two different input vectors having the same image.

• Is this transformation onto? (Hint: Review problem#6 of Homework 06.)
  1. If so, explain which properties of the matrix make the transformation onto.
  2. If not, given an example of a vector in R^m that is not the image of any vector in Rⁿ.