Chapter 10, Change of bases Video Solutions, Linear Algebra

Problem 1

Consider $\mathbb{R}^{3}$ with two orthonormal bases: the canonical basis $e=\left(e_{1}, e_{2}, e_{3}\right)$ and the basis $f=\left(f_{1}, f_{2}, f_{3}\right)$, where
$$
f_{1}=\frac{1}{\sqrt{3}}(1,1,1), f_{2}=\frac{1}{\sqrt{6}}(1,-2,1), f_{3}=\frac{1}{\sqrt{2}}(1,0,-1)
$$
Find the matrix, $S$, of the change of basis transformation such that
$$
[v]_{f}=S[v]_{e}, \text { for all } \mathrm{v} \in \mathbb{R}^{3}
$$
where $[v]_{b}$ denotes the column vector of $v$ with respect to the basis $b$.

Victor Salazar

Numerade Educator

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Problem 2

Let $v \in \mathbb{C}^{4}$ be the vector given by $v=(1, i,-1,-i)$. Find the matrix (with respect to the canonical basis on $\mathbb{C}^{4}$ ) of the orthogonal projection $P \in \mathcal{L}\left(\mathbb{C}^{4}\right)$ such that
$$
\operatorname{null}(P)=v^{\perp}
$$

Nick Johnson

Numerade Educator

07:38

Problem 3

Let $U$ be the subspace of $\mathbb{R}^{3}$ that coincides with the plane through the origin that is perpendicular to the vector $n=(1,1,1) \in \mathbb{R}^{3}$
(a) Find an orthonormal basis for $U$.
(b) Find the matrix (with respect to the canonical basis on $\mathbb{R}^{3}$ ) of the orthogonal projection $P \in \mathcal{L}\left(\mathbb{R}^{3}\right)$ onto $U$, i.e., such that $\operatorname{range}(P)=U$

Chris Trentman

Numerade Educator

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Problem 4

Let $V=\mathbb{C}^{4}$ with its standard inner product. For $\theta \in \mathbb{R}$, let
$$
v_{\theta}=\left(\begin{array}{c}
1 \\
e^{i \theta} \\
e^{2 i \theta} \\
e^{3 i \theta}
\end{array}\right) \in \mathbb{C}^{4}
$$
Find the canonical matrix of the orthogonal projection onto the subspace $v_{\theta} \perp$.

Nick Johnson

Numerade Educator