I am currently stuck on number 16:
So theorem 4.5 says:
If $W$ is a non empty subset of a vector space $V$, then $W$ is a subspace of $V$ if and only if the following closure conditions hold:
- If $\mathbf{u}$ and $\mathbf{v}$ are in (subspace) $W$ then $\mathbf{u} + \mathbf{v}$ is in $W$.
- If $\mathbf{u}$ is in $W$ and $c$ is any scalar, then $c\mathbf{u}$ is in $W$.
Questions:
1) What is defined as a closure?
2) I tried doing 16 by saying that A is the 2x2 matrix $\left(\begin{array}{cc}1&0\\0&0\end{array}\right)$, and that B is the 2x2 matrix $\left(\begin{array}{cc}0&0\\0&1\end{array}\right)$. Both of these are singular. So if you add the two together you get $\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$ which is nonsingular. Thus we can conclude that it is not closed under addition. So no.1 is not satisfied.
3) How do I know if condition 2 is satisfied? If I multiply some scalar, such as 2, by matrix A from 2), I would get $\left(\begin{array}{cc}2&0\\0&0\end{array}\right)$. Is this considered to be part of the subspace?