MA 2071 Abstract Vector Spaces

M₂₂ = set of all 2x2 matrices

P₂ = set of all 2nd degree polynomials

Note: you must either prove that it is a vector space using the definition or come up with a counterexample to disprove it

1. The set of matrices where a₁₂ = - a₂₁ in M₂₂

2. The set of all upper triangular matrices in M₃₃

3. { x² + bx + c } in P₂

4. { all "even" functions; f(x) = f(-x) }

5. { all solutions to d²y/dx² + y = 0 }

6. {all f(x) such that integral a to b of f(x)dx = 0 } (a and b fixed )

7. {all nonsingular (invertible) 3x3 matrices }

8. { matrices with 0 on the diagonal } in M₂₂

9. { all f(x) where f(0) = 1 }

10. { ax² + bx + c where a = 2b} in P₂

11. In problems 1,2 and 7 can you find a) a basis ? b) tell what the dimension is?