A vector space [tex]V[/tex] is a subspace of a vector space [tex]W[/tex] if
It's easy to show the first condition is met by all the sets in parts (a-g).
(a) is a subspace of [tex]\Bbb R^{2\times2}[/tex] because adding any 2x2 diagonal matrices together, or multiplying one by some scalar, gives another diagonal matrix.
(b) and (c) are also subspaces for the same reasons.
(d) is not a subspace because [tex]\Bbb R^{2\times2}[/tex] because this set of matrices does not contain the zero matrix.
(e), however, is a subspace. Any linear combination of matrices in this set always yields a matrix with 0 in row 1, column 1 entry.
(f) is a subspace. A symmetric matrix is one of the form
[tex]\begin{bmatrix}a&b\\b&c\end{bmatrix}[/tex]
Adding two symmetric matrices gives another symmetric matrix:
[tex]\begin{bmatrix}a_1&b_1\\b_1&c_1\end{bmatrix}+\begin{bmatrix}a_2&b_2\\b_2&c_2\end{bmatrix}=\begin{bmatrix}a_1+a_2&b_1+b_2\\b_1+b_2&c_1+c_2\end{bmatrix}[/tex]
(g) is not a subspace. Consider the matrices
[tex]\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}[/tex]
Both matrices have determinant 0, but their sum is the identity matrix with determinant 1.
