Homework 8 - Inner products and Gram Schmidt
Due 2021-10-19.
Let $\langle -,- \rangle$ be a bilinear pairing on a vector space $V$. Let $W \subseteq V$ be a subspace. We define the orthogonal to be the subset
\(W^{\perp} = \lbrace v \in V \mid \langle v,w \rangle = 0 ~\forall w \in W \rbrace\) Show that $W^{\perp}$ is also a subspace of $V$.Continuing with the set up from Problem 1. If $\langle -,- \rangle$ is an inner product and $V$ is finite dimensional, show that \(W = \left( W^{\perp} \right)^{\perp}\)
Perform Gram Schmidt on the collection of vectors \(\begin{pmatrix} 1 \\ 2 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -1 \\ 1 \end{pmatrix}\)
Let $V$ be a finite dimensional vector space with a positive definite inner product, $U \subseteq V$ a subspace, and $v \in V$. The projection of $v$ onto $U$ is the vector \(\operatorname{proj}_U(v) = \sum_{s=1}^d \langle v,u_s \rangle u_s\) where $u_1,\ldots,u_d$ is an orthonormal basis for the subspace $U$.
Show that \(v - \operatorname{proj}_{U}(v) = \operatorname{proj}_{U^{\perp}}(v)\)
(Hint: what is $\operatorname{proj}_V(v)$? Using a orthonormal basis for $V$ that splits up into a union of one for $U$ and one for $U^\perp$.)
Show that $\operatorname{proj}_U(v)$ does not depend on the choice of orthonormal basis of $U$ used.