Question

We have learned that we can consider spaces of matrices, polynomials or functions as vector spaces. For the following examples, use the definition of subspace to determine whether the set in question is a subspace or not (for the given vector space), and why.

1. The set M1 of 2×2 matrices with real entries such that all entries of their diagonal are equal. That is, all 2 × 2 matrices of the form: A = a b c a

2. The set M2 of 3×3 matrices with real entries such that the entry A(1, 3) = 1.

3. The set Q of quadratic polynomials p(x) = ax2+bx+c such that p 0 (2) = 0.

4. The set C of cubic polynomials p(x) = ax3+bx2+cx+d such that p(3) = 7 and p(2) = 1. Remember: You can first check whether the vector 0 is in the set for a quick check. If it is, you still need to check whether sums and product by a scalar takes you out of the set (or not).

Answer 1