Orthogonality

Let $A=\left[\begin{array}{ccc}1&1&0\\ 1&2&0\\ 0&0&2\end{array}\right]$, and consider $\mathbb{R}^{3}$ with the inner product given by $\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}^{T}A\mathbf{v}$. Which of the following vectors are orthogonal to each other?

On $C[-1,1]$, which of the following functions are orthogonal to each other?

Consider the inner product space $\mathbf{P}_{3}$ of polynomials of degree at most $3$, with the inner product defined by

1. Find the orthogonal complement of $\left\{x^{2},x\right\}$.

2. Find the orthogonal complement of $\left\{x+1\right\}$.

Consider $\mathbb{R}^{3}$ as an inner product space with the usual dot product. For each of the following bases of $\mathbb{R}^{3}$, state whether it is orthonormal, orthogonal, or neither.

1. $\left\{\left[\begin{array}{r}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]\right\}$.

2. $\left\{\left[\begin{array}{r}1\\ 0\\ 1\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]\right\}$.

3. $\left\{\left[\begin{array}{r}1\\ 0\\ 2\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}-2\\ 0\\ 1\end{array}\right]\right\}$.

4. $\left\{\left[\begin{array}{r}\frac{3}{5}\\ \frac{4}{5}\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 0\\ -1\end{array}\right],\left[\begin{array}{r}\frac{4}{5}\\ -\frac{3}{5}\\ 0\end{array}\right]\right\}$.

Suppose $B=\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$ is an orthogonal basis for an inner product space $V$, such that $\left\|\mathbf{u}_{1}\right\|=2$, $\left\|\mathbf{u}_{2}\right\|=\sqrt{3}$, and $\left\|\mathbf{u}_{3}\right\|=\sqrt{5}$. Moreover, suppose that $\mathbf{v}\in V$ is a vector such that $\langle\mathbf{v},\mathbf{u}_{1}\rangle=1$, $\langle\mathbf{v},\mathbf{u}_{2}\rangle=2$, and $\langle\mathbf{v},\mathbf{u}_{3}\rangle=-4$. Find the coordinates of $\mathbf{v}$ with respect to $B$.

Suppose $B=\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$ is an orthogonal basis of $\mathbb{R}^{3}$. We have been told that

but it is not known what $\mathbf{u}_{2}$ and $\mathbf{u}_{3}$ are. Find the first coordinate of the vector

with respect to the basis $B$.