Linear Algebra: vectors and orthonormal basis

Question

1. $\text{(a)}$ Let $\mathbf{u}=[1\,\,2\,\,3\,\,1]^T$ and $\mathbf v=[0\,\,0\,\,1\,\,-3]^T$ in $\mathbb{R}^4$. Find an orthonormal basis for the space $\mathcal W$ such that $\mathbf w \,\bot\, \mathbf u$ and $\mathbf w \,\bot\, \mathbf v$ for all $\mathbf w \in \mathcal W$.

$\quad\,\,\,\,\text{(b)}$ Let $\{\mathbf v_1,\,\mathbf v_2,\ldots,\mathbf v_n\}$ be an orthonormal basis of an $n$ dimensional space $\mathcal V$. For any $\quad\,\,\,\,$vector $\mathbf y \in\mathcal V$, prove that $$\parallel\mathbf y\parallel^2\ge\sum_{i=1}^rc_ i^2$$ $\quad\,\,\,\,$with $r\le n$ and $c_i$ is the orthogonal projection coefficient of $\mathbf y$ onto $\mathbf v_i$.

For $\text{(a)}$, is that I was asked to find w and show that u.w = v.w = 0 ?

For $\text{(b)}$, would anyone mind giving hints to me or just telling me what I should do so as to solve this problem?

Answer 1

For (a): yes. There is a very obvious choice for $w$, hint: $w=(?,?,0,0)^T$. Now normalize $u$, $v$ and $w$.

For (b): do you know how to compute $c_k$?

Answer 2

(a) $\mathcal W$ is a 2-dimensional linear space and you are asked to find an orthonormal basis for it. You need to find two vectors $\textbf{w}_1$ and $\textbf{w}_2$ such that $\textbf{w}_1\cdot\textbf{u}=0$, $\textbf{w}_1\cdot\textbf{v}=0$, $\textbf{w}_2\cdot\textbf{u}=0$, $\textbf{w}_2\cdot\textbf{v}=0$, $\textbf{w}_1\cdot\textbf{w}_2=0$, and $||\textbf{w}_1||=||\textbf{w}_2||=1$ . One of these vectors is pretty obvious, $\textbf{w}_1=\frac{1}{\sqrt{5}}[-2\ \ 1\ \ 0\ \ 0]^T$ (assuming Euclidean metric), and you are left with the four equations for the coordinates of $\textbf{w}_2$.

(b) $\textbf{y}=\sum_{i=1}^n c_i\textbf{v}_i$ and $||\textbf{y}||^2=\sum_{i=1}^n c_i^2$ if the space is Euclidean, which implies the inequality $||\textbf{y}||^2\ge\sum_{i=1}^r c_i^2$ for $r\le n$.