MA 2071 ** Abstract Vector Spaces**

M_{22} = set of all 2x2 matrices

P_{2} = 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** a _{12} = - a_{21}** in M

2. The set of all upper triangular matrices in M_{33}

3. { x^{2} + bx + c } in P_{2}

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

5. { all solutions to d^{2}y/dx^{2} + 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 M_{22}

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

10. { ax^{2} + bx + c where a = 2b} in P_{2}

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