1. Linear Algebra Help!!!

I am struggling with a few homework problems.

1. a) Show that the set of nonsingular 2x2 matrices is not a subspace of M2x2.

b) Show that the set of singular 2x2 matrices is not a subspace of M2x2.

(Let M2x2 be the vector space of all 2x2 matrices for both)

2. Describe the smallest subspace of M2x2 that contains
(1 0) (1 0)
(0 0) and (0 1) then find dimension of the subspace.

3. Let B = {1 + t - 2t^2, t - t^2, 2 - 2t + t^2}. Check that B is a basis for P2 and find [3 + t - 6t^2]b.

2. Originally Posted by jkong
I am struggling with a few homework problems.

1. a) Show that the set of nonsingular 2x2 matrices is not a subspace of M2x2.

b) Show that the set of singular 2x2 matrices is not a subspace of M2x2.

(Let M2x2 be the vector space of all 2x2 matrices for both)

2. Describe the smallest subspace of M2x2 that contains
(1 0) (1 0)
(0 0) and (0 1) then find dimension of the subspace.

3. Let B = {1 + t - 2t^2, t - t^2, 2 - 2t + t^2}. Check that B is a basis for P2 and find [3 + t - 6t^2]b.
Q1. Apply the subspace theorem. Can you find elements u and v such that

a) u and v are nonsingular but $\alpha u + \beta v$ is NOT nonsingular?

b) u and v are singular but $\alpha u + \beta v$ is NOT singular?

Q2. Due to formatting ambiguities I can't tell what the two given matrices are.

Q3. What's P2? The set of second degree polynomials? If so, check whether:

1. B spans (that is, at^2 + bt + c can be constructed from a linear combination of the elements of B).
2. The elements of B are linearly independent.

For the second part, it's not clear to me what you're trying to find.