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1) The set of all polynomials degree two.

2) The set of all diagonal 2 x 2 matrices.

For the first one, it will not be a vector space because it does not satisfy the closure property. Also the distributive property would be broken because a(x+y)^2 would not be (ax+ay)^2, or did I do that wrong?

The second one would not satisfy additivity properties, and not be a vector space.

Right? I don't know the set notation in LaTeX, so I'm not really sure how to put up much of my work.