Question 7 [10 points] Let S be the transformation whose matrix is A, and let T
ID: 3116850 • Letter: Q
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Question 7 [10 points] Let S be the transformation whose matrix is A, and let T be the transformation whose matrix is B, where A and B are the matrices below. Find the matrix C for the transformation resulting from S followed by T. 10 10102 5 C- 0 0 0 Question 8 [10 points] Suppose T is a transformation from R2 to R2. Find the matrix A that induces Tif T IS a) rotation by 1 b) reflection over the line y=54x a)A=100 b)A 0 0 Question 9 [10 points] Suppose TR3R4 is the transformation induced by the following matrix A Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two vectors that have the same image under T. If T is not onto, show this by providing a vector in R4 that is not in the range of T. 5-10 -5 2-3-3 1 02 2 -3-1 T is one-to-one T is ontoExplanation / Answer
7. The matrix of the transformation SoT is BA =
92
-103
-101
66
-44
-48
-52
-47
-29
8.a). If T is a counterclockwise rotation, then its matrix representation is
cos
-sin
sin
cos
=
-1
0
0
-1
If T is a clockwise rotation, then its matrix representation is
cos
sin
-sin
cos
=
-1
0
0
-1
b). The matrix representation of a transformation for reflection across the line y = mx is
(1-m2)/(1+m2)
2m/(1+m2)
2m/(1+m2)
-(1-m2)/(1+m2)
Here, m = 5/4, so that the required matrix is
-9/41
40/41
40/41
9/41
9. To determine whether T is one-to-one or onto, we will reduce A to its RREF as under:
Multiply the 1st row by 1/5
Add -2 times the 1st row to the 2nd row
Add -1 times the 1st row to the 3rd row
Add -2 times the 1st row to the 4th row
Add -2 times the 2nd row to the 3rd row
Add -1 times the 2nd row to the 4th row
Add -2 times the 3rd row to the 4th row
Add 1 times the 3rd row to the 2nd row
Add 1 times the 3rd row to the 1st row
Add 2 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
0
1
0
0
0
1
0
0
0
It may be observed that the columns of A are linearly independent. Hence T is one-to-one. Further, the columns of A do not span R4 as the last row of the RREF of A has only zeros. Therefore, T is not onto. Any vector of the form(a,b,c,d)T where d0, does not have a pre-image under T in R3.
92
-103
-101
66
-44
-48
-52
-47
-29
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