How do I show that this is a direct sum?

Question

$U$ and $V$ are vector spaces.

$m_1 = \begin{pmatrix}1&0\\0&0\end{pmatrix}$, $m_2 = \begin{pmatrix}0&0\\0&1\end{pmatrix}$, $m_3 = \begin{pmatrix}0&1\\1&0\end{pmatrix}$ is a basis of $U$.

And:

$p_1 = \begin{pmatrix}0&1\\-1&0\end{pmatrix}$ is a basis of $V$.

How do I show that $U + V$ is a direct sum?

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I'm learning linear algebra, so I want to apologize for any mistakes.

Also, just got into math.stackexchange, so I'm still picking up the formatting.

Thanks in advance.

Answer 1

Let $q \in U \cap V$, then there are scalars $t_1,...,t_4$ such that

$$q=t_1m_1+t_2m_2+t_3m_3=t_4p_1.$$

This gives a system of four linear equations for $t_1,...,t_4$ .

Solve this system and look what happens....

Answer 2

Observe that $$V=\left\{\begin{bmatrix}0&b\\-b&0\end{bmatrix} \, \Big| \, b \in \Bbb{R}\right\}$$ Likewise, $$U=\left\{\begin{bmatrix}p&r\\r&q\end{bmatrix} \Big| p,q,r \in \Bbb{R}\right\}.$$ So the only common element among them is the zero matrix (because we would require $p=q=0$ and $r+r=0$), hence $U \cap V=\{\mathbf{0}\}$.