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Math 205A Test 2 (50 points)
Name:
- Check that you have 6 questions on two pages.
- Show all your work to receive full credit for a problem.
- (12 points) Short answers: (No explanations needed. Simply write your answers. If you do some calculation to get the answer, show the calculation.)
(a) Find the characteristic polynomial of the matrix
[
]
. (Do not solve the charac-
teristic equation to find the eigenvalues; simply find the polynomial.)
(b) Find the distance between the vector ~u =
and the vector^ ~v^ =
(c) Find the orthogonal projection of the vector ~y =
(^) onto the vector ~u =
(d) Let ~u 1 =
(^) and let ~u 2 =
. Let W = Span {~u 1 , ~u 2 }. Then the set B = {~u 1 , ~u 2 }
is a basis for W. For the vector ~y ==
(^) in W , find [~y]B.
(e) For a 5 × 8 matrix A, Col A = R^5. What is the dimension of Nul A?
- (6 points) Let A and B be 3 × 3 matrices, with det A = 2 and det B = −5. Use properties of determinants to compute:
(a) det 10A
(b) det BT^ A
(c) det (BA)−^1 if BA is invertible. Otherwise, explain why BA is not invertible.
- (6 points) Let W = Span
{[
]}
(a) Draw W and W ⊥.
(b) Find a spanning set for W ⊥.
- (8 points) Let W be a subspace of R^5 and let B = {~v 1 , ~v 2 } be an orthogonal basis for W.
(a) Find dim W.
(b) Is the set { 2 ~v 1 , − 3 ~v 2 } an orthogonal set? Explain.
(c) Is the set { 2 ~v 1 , − 3 ~v 2 } a basis for W? Use the two conditions in the definition of a basis to explain your answer.
