1. a) Find the image of the unit triangle POQ, where O = (0,0), Q = (1,0) and P = (0,1), upon
transformation by the matrix
A=
3 0
0 1
b) What is this type of transformation known as?
c) Draw the triangle POQ and its image P’O’Q’ by scatter chart of Excel.
Verify your work numerically: take a point R on PQ, such as R (0.8,0.2). Compute the image of R
under A, find the equation of P’Q’ and verify that the image of R indeed lies on P’Q’.
2. What are the matrices for the following geometric linear transformations?
a) Reflection through the line x2 = -x1
b) Horizontal shear by 2.5 units
c) (a) followed by (b) (one matrix)
3. Find the determinant of the following matrix:
A =
1 −2 −1
−1 5
6
5 −4 5
Is A invertible? Use the determinant to answer this question – do not compute or use Excel.
4. Repeat Q3 on the following matrix:
1
0 −2
B = −3 1
4
2 −3 4
If B is invertible, use Excel to find the inverse. Verify your result by checking that BB-1 = I.
5. Determine if the following vectors are linearly independent. Justify your answer.
7
2,
−6
5
0,
0
6. Is 𝜆 = 2 an eigenvalue of
3
3
9
4
−8
2
? Why or why not?
8
4
3
7 9
7. Is −3 an eigenvector of −4 −5 1 ? If so, find the eigenvalue.
1
2
4 4
8. Let 𝜆 be an eigenvalue of an invertible matrix A. Prove that 𝜆-1 is an eigenvalue of A-1.
9. Find the characteristic equation, eigenvalues and corresponding eigenvectors of the matrix:
A =
4 −5
2 −3
Then plot the eigenvectors and their images under A. What do you observe?
For each eigenvector, compute the angle with its image under A.
10. Let u and v be the vectors as shown in the figure, and support u and v are eigenvectors of a 2 X 2
matrix A that correspond to eigenvalues 2 and 3, respectively. Let w = u + v. Make a copy of the
figure, and on the same coordinate system, carefully plot the vectors Au, Av and Aw.
x2
v
u
x1