Name:
CSU ID:
Homework 7
October 23, 2015
1. Find the standard matrix for the stated composition in <3
(a) The rotation of 30◦ about the x-axis, followed by a rotation of 30◦
about the z-axis, followed by a contraction with factor k = 1/4.
(b) A reflection about the xy-plane followed by a reflection about the
xz-plane, followed by an orthogonal projection onto the yz-plane.
2. Let B = {M1 , M2 , M3 , M4 } be the ordered basis given below. Find the
coorinates of the vector M = I2×2 relative to the ordered basis B.
"
M1 =
1 0
2 0
#
"
, M2 =
−1 5
0 2
#
"
, M3 =
4 6
8 3
#
"
, M4 =
3 −4
6
3
#
3. Let B = 1 + 2x2 , −1 + 5x + 2x3 , 4 + 6x + 8x2 + 3x3 , 3 − 4x + 6x2 + 3x3 .
Show that B is a basis for P 3 , polynomials of degree less than or equal
to 3. For the polynomial p(x) = 1 + x + x2 + x3 , determine [p]B .
6. Find the eigenvalues and eigenvectors of A where


1 3
0


0 
A= 4 1
0 0 −2
7. Find the eigenvalues and eigenvectors of A


0 −3
5


4 −10 
A =  −4
0
0
4
8. The matrix A given below has eigenvalues λ = 2, −5, 3.
(a) For each eigenvalue, find the eigenvector by bringing λI − A to
RREF. The eigenvector should be defined with integers.
(b) Define a matrices S and D such that S −1 AS = D is a diagonal
matrix with the ordered eigenvalues given in (a). Do Not Solve
for S −1


5 −4 −2


27
14 
A =  −2
5 −50 −26
9. Show that for A in the previous problem that the characteristic polynomial of A, pA (x), satisfies pA (A) = 03×3