Determine whether the following subsets of V are subspaces of V. Justify your answers: if the subset is a subspace, verify the necessary properties a subspace must have; if it is not a subspace, show at least one required property which fails.
$W=$ set of invertible $2\times2$ matrices in $V=M_{22}$
I did the following test using a specific example:
$A=$$\begin{bmatrix} 1 & 0 \\ 0 & 1
\\ \end{bmatrix} +$ $B=$$\begin{bmatrix} 1 & 0 \\ 0 & 1
\\ \end{bmatrix} $ $=$$\begin{bmatrix} 2 & 0 \\ 0 & 2
\\ \end{bmatrix}$
This is a vector space in the set, but how do I know all invertible matrices fall into this category?
Best Answer
$W$ is not closed under vector addition (and any subspace of a vector space must be closed under vector addition). Hence $W$ cannot be a subspace of $V$.
How do we know $W$ is not closed under addition? To show this we provide a counter-example:
Let $A\in W$. Then $-A \in W$. (I.e., if $A$ is invertible, $-A$ is invertible).
But $A + (-A) = {\bf {0}} \notin W$, since $\bf 0$ (the zero vector) is not invertible (and $W$ is defined to be the set of all invertible $2\times 2$ matrices).
Hence our conclusion that $W$ is not closed under vector addition.
One can also simply argue that $W$ fails to contain the zero matrix, by definition of $W$. And since every vector subspace (space) must contain the zero vector, $W$ fails to be a subspace of $V$.