# The Unapologetic Mathematician

## Orthogonal Complements

An important fact about the category of vector spaces is that all exact sequences split. That is, if we have a short exact sequence

$\displaystyle\mathbf{0}\rightarrow U\rightarrow V\rightarrow W\rightarrow\mathbf{0}$

we can find a linear map from $W$ to $V$ which lets us view it as a subspace of $V$, and we can write $V\cong U\oplus W$. When we have an inner product around and $V$ is finite-dimensional, we can do this canonically.

What we’ll do is define the orthogonal complement of $U\subseteq V$ to be the vector space

$\displaystyle U^\perp=\left\{v\in V\vert\forall u\in U,\langle u,v\rangle=0\right\}$

That is, $U^\perp$ consists of all vectors in $V$ perpendicular to every vector in $U$.

First, we should check that this is indeed a subspace. If we have vectors $v,w\in U^\perp$, scalars $a,b$, and a vector $u\in U$, then we can check

$\displaystyle\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle=0$

and thus the linear combination $av+bw$ is also in $U^\perp$.

Now to see that $U\oplus U^\perp\cong V$, take an orthonormal basis $\left\{e_i\right\}_{i=1}^n$ for $U\subseteq V$. Then we can expand it to an orthonormal basis $\left\{e_i\right\}_{i=1}^d$ of $V$. But now I say that $\left\{e_i\right\}_{i=n+1}^d$ is a basis for $U^\perp$. Clearly they’re linearly independent, so we just have to verify that their span is exactly $U^\perp$.

First, we can check that $e_k\in U^\perp$ for any $k$ between $n+1$ and $d$, and so their span is contained in $U^\perp$. Indeed, if $u=u^ie_i$ is a vector in $U$, then we can calculate the inner product

$\displaystyle\langle u^ie_i,e_k\rangle=\bar{u^i}\langle e_i,e_k\rangle=\bar{u^i}\delta_{ik}=0$

since $i\leq n$ and $k\geq n+1$. Of course, we omit the conjugation when working over $\mathbb{R}$.

Now, let’s say we have a vector $v\in U^\perp\subseteq V$. We can write it in terms of the full basis $\left\{e_k\right\}_{k=1}^d$ as $v^ke_k$. Then we can calculate its inner product with each of the basis vectors of $U$ as

$\displaystyle\langle e_i,v^ke_k\rangle=v^k\langle e_i,e_k\rangle=v^k\delta_{ik}=v^i$

Since this must be zero, we find that the coefficient $v^i$ of $e_i$ must be zero for all $i$ between ${1}$ and $n$. That is, $U^\perp$ is contained within the span of $\left\{e_i\right\}_{i=n+1}^d$

So between a basis for $U$ and a basis for $U^\perp$ we have a basis for $V$ with no overlap, we can write any vector $v\in V$ uniquely as the sum of one vector from $U$ and one from $U^\perp$, and so we have a direct sum decomposition as desired.