1. Verify that functions defined by a matrix is always linear. More precisely, verify that LA : R2 → R2, LA(~x) = A~x, with , is linear.
2. Determine whether each of the following functions is linear or not. Explain your reasoning.
3. 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 = LA.
(a) T stretches by a factor of 2 in the x-direction and by a factor of 3 in the y-direction.
(b) T is a reflection across the line y = x.
(c) T is a rotation (about the origin) through −π/4 radians.
(d) T is a vertical shear that maps ~e1 into ~e1−~e2 but leaves the vector ~e2 unchanged.
4. For any given m×n matrix A, we are going to use the notation LA to denote the linear transformation that A defines, i.e., LA : Rn→Rm : LA(~x) = A~x. For each given matrix, answer the following questions.
(a) Rewrite LD : Rn→Rm with correct numbers for m and n filled in for each matrix. Repeat for LE and LF.
(b) Find some way to explain in words and/or graphically what this transformation does in taking vectors from Rn to Rm. You might find it helpful to try out a few input vectors and see what their image is under the transformation.
(c) Is this transformation one-to-one? (Hint: Review problem#6 of Homework 06.)
i. If so, explain which properties of the matrix make the transformation one-to-one.
MATH 141: Linear Analysis I Homework 07 Fall 2019
ii. If not, given an example of two different input vectors having the same image.
(d) Is this transformation onto?
i. If so, explain which properties of the matrix make the transformation onto.
ii. If not, given an example of a vector in Rm that is not the image of any vector in Rn.
