linear algebra
steve912
e5kvjF-242h4.pdf
MAT 242 Written Homework #4
EP4.4, 7.3, 6.1, 6.2 / H4.4, 5.1, 5.2, 5.A Due: November 2
Solve the following problems, showing any necessary work.
1. [3 points] For the matrix A =
0 4 −8 0 8 0 1 4 −13 −3 6 −3
−3 2 11 9 10 −2 −2 3 4 6 10 4 −2 1 8 6 6 4 −2 1 8 6 6 1
, find a basis for the null space of A, a basis
for the row space of A, a basis for the column space of A, the rank of A, and the nullity of A.
2. Let B be the (ordered) basis
11
2
,
−2−1 −5
,
−30 −8
and C the basis
11
2
,
−10 −2
,
31
7
.
a. [5 points] Find the coordinates of
125
23
with respect to the basis C.
b. [10 points] Find the change-of-basis matrix from C to B.
3. Let A =
[ −4 4 −4 0 −2 0 2 −5 2
] .
a. [1 point] Find the eigenvalues of A.
b. [1 point] Find a basis for the eigenspace of each eigenvalue, indicating which is which.
c. [1 point] Is the matrix A diagonalizable? If so, find matrices D and P such that A = PDP −1 and D is a diagonal matrix. If A is not diagonalizable explain carefully why it is not diagonalizable.
