Euclidean space and set $M\cap M^{\bot}$

Question

If $M$ is subspace of some Euclidean space $E$ than set $M\cap M^{\bot}$ sometimes is empty, sometimes is not.

I read in book that $M\cap M^{\bot}=\{0\}$ so it can not be sometime empty sometimes not, is this true?

Answer 1

Note, that every subspace of a vector space $V$ contains the null vector, i.e. the intersection of subspaces is never empty.

Now, let $(V,\langle\cdot,\cdot\rangle)$ be a euclidean space. Note also, that $M\bot M^\bot$, i.e. that $M$ and its orthogonal complement are orthogonal.

To see this, let $m\in M, m'\in M^\bot$. Now, $\langle m,m'\rangle=0$ as

$$M^\bot=\{v\in V\mid\forall m\in M:\langle m,v\rangle =0\}$$

i.e. as $m'\in M^\bot$, it is specifically orthogonal to $m\in M$.

Now, let $v\in M\cap M^\bot$, i.e. as $M\bot M^\bot$, we have $\langle v,v\rangle=0$ by the above as $v$ is in both subspaces and they are orthogonal, i.e. $v=\mathbf0$ by positive definiteness of the inner product $\langle\cdot,\cdot\rangle$.