Exercise 13

One advantage of orthogonal bases is that they simplify finding the representation of a vector with respect to that basis.

• For this vector and this non-orthogonal basis for $\mathbb{R}^{2}$

  $$\vec{v}=\begin{pmatrix}2\\ 3\end{pmatrix}\qquad B=\langle\begin{pmatrix}1\\ 1\end{pmatrix},\begin{pmatrix}1\\ 0\end{pmatrix}\rangle$$

  first represent the vector with respect to the basis. Then project the vector into the span of each basis vector $[\smash{\vec{\beta}_{1}}]$ and $[\smash{\vec{\beta}_{2}}]$.

• With this orthogonal basis for $\mathbb{R}^{2}$

  $$K=\langle\begin{pmatrix}1\\ 1\end{pmatrix},\begin{pmatrix}1\\ -1\end{pmatrix}\rangle$$

  represent the same vector $\vec{v}$ with respect to the basis. Then project the vector into the span of each basis vector. Note that the coefficients in the representation and the projection are the same.

• Let $K=\langle\vec{\kappa}_{1},\ldots,\vec{\kappa}_{k}\rangle$ be an orthogonal basis for some subspace of $\mathbb{R}^{n}$. Prove that for any $\vec{v}$ in the subspace, the $i$-th component of the representation ${\rm Rep}_{K}(\vec{v}\,)$ is the scalar coefficient $(\vec{v}\mathbin{\smash{\bullet}}\vec{\kappa}_{i})/(\vec{\kappa}_{i}\mathbin{\smash{\bullet}}\vec{\kappa}_{i})$ from $\text{proj}_{[\vec{\kappa}_{i}]}({\vec{v}\,})$.

• Prove that $\vec{v}=\text{proj}_{[\vec{\kappa}_{1}]}({\vec{v}\,})+\dots+\text{proj}_{[\vec{\kappa}_{k}]}({\vec{v}\,})$.