How to show if the following subset $W$ is a subspace of a vector space $V$?

Question

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$

$2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$

$3.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R})\mid A \text{ is invertible}\}$

For the first one, I think let $p,q \in W$ and $c\in R$, then $(cp+q)(1)+(cp+q)(2)+(cp+q)(3)=c(p(1)+p(2)+p(3))+q(1)+q(2)+q(3)=0$. Therefore it is subspace of $V$.

For the second one, I think not symmetric can be represented as $a_{ij}\ne a_{ji} $ for some $j, i$. Then let $A, B\in W$ and $c\in R$, $ca_{ij}+b_{ij}-ca_{ji}+b_{ji}\ne 0$, $c(a_{ij}-a_{ji})+b_{ij}-b_{ji}\ne 0$. Here I saw a contradiction; what if $c$ is $0$?

For the last one, It can be written as $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n} (\mathbb{R}) \mid \det{A}\ne 0\}$. But how do you make it more explicit, as it's hard to prove this way?

Could anyone please correct my procedure?

Answer 1

Your proof for the first one is correct.

For the second one, look at $\left(\begin{matrix} 0 & 1\\ 0 & 0 \end{matrix}\right)+\left(\begin{matrix} 0 & 0\\ 1 & 0 \end{matrix}\right)$.

For the third, look at $\left(\begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right)+\left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right)$.

Answer 2

Hints/Comments: 1. Correct 2. Also correct but it would be clearer if you gave an explicit counterexample. Can you think of two matrices $A$ and $B$ that are not symmetric but for which $A+B$ is symmetric? Don't overthink this one! 3. Can you think of two invertible matrices $A$ and $B$ whose sum is not invertible? Try simple examples in which the determinants of $A$ and $B$ are easy to calculate.

In both 2. and 3. it will help to start with $n = 2$ and then generalize. Good luck!