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MATH 141: Linear Analysis I Homework 07 solved

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1. Verify that functions defined by a matrix is always linear. More precisely, verify that LA : R
2 → R
2
,
LA(~x) = A~x, with A =

a b
c d
, is linear.
2. Determine whether each of the following functions is linear or not. Explain your reasoning.
(a) T : R → R, T(x) = x
2
.
(b) T : R → R, T(x) = x + 3.
(c) T : R
2 → R
2
, T
x1
x2
 =

3×1
x1 + 2×2

(d) T : R
2 → R
2
, T
x1
x2
 =

x1
2×2 − 1

3. Assume that T : R
2 → R
2
is a linear transformation. Let ~e1 =

1
0

and ~e2 =

0
1

. 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.
© IOLA Team – iola.math.vt.edu
7. Consider the image given below and the transformation matrix ! = 2 0
0 −1.5
a. Sketch what will happen to the image under the transformation.
b. Describe in words what will happen to the image under the transformation.
c. Describe how you determined that happened. (What, if any, calculations did you do?
Did you make a prediction? How did you know you were right? etc.
8. Assume that T is a linear transformation and that !! = 1
0 and !! = 0
1 . For each part, find the standard
matrix A for T, and draw the image of the “half-shaded unit square” (shown below) under the given
transformation.
a. !: ℝ! → ℝ! rotates points (about the origin) through –π/4 radians (clockwise)
b. !: ℝ! → ℝ! is a vertical shear that maps e1 into e1 – e2 but leaves the vector e2 unchanged
c. !: ℝ! → ℝ! first reflects points across the vertical axis and then rotates points π/2 radians
(counterclockwise)
C
?
A
(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 : R
n → R
m : LA(~x) = A~x. For each given matrix, answer the following questions.
D =


3 0 0
0 −1 0
0 0 1/2

 E =


4 0
0 0
0 2

 F =

2 0 1
0 3 0
(a) Rewrite LD : R
n → R
m 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 R
n to R
m. 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
ii. If not, given an example of two different input vectors having the same image.
(d) Is this transformation onto? (Hint: Review problem#6 of Homework 06.)
i. If so, explain which properties of the matrix make the transformation onto.
ii. If not, given an example of a vector in R
m that is not the image of any vector in R
n.