# Harrison Chapman

## Linear Algebra I: Homework 10

Due: Friday, November 10, 2017
1. For vectors $\vec v = \begin{pmatrix}v^1 \\ v^2\end{pmatrix}$ and $\vec w = \begin{pmatrix}w^1 \\ w^2\end{pmatrix}$, $\langle \vec v, \vec w \rangle = 3v^1w^1 + 2v^2w^2$ is an inner product.
1. Find all unit vectors in $\mathbb R^2$ with respect to this new inner product.

See Homework 7.2.a; this is similar but with a different magnitude. Unit vectors are the ellipse determined by $1 = 3x^2 + 2y^2$, that is, the set of vectors;

2. Find two different orthonormal bases for $\mathbb R^2$ with respect to this new inner product.

Our solution for (a) says that two vectors determined by angles $\alpha$ and $\beta$ have dot product,

This means that two of our unit vectors are orthogonal if $\alpha$ and $\beta$ form a right angle (why?). So to come up with orthonormal bases we just have to pick an angle, and then another angle which is orthogonal to it:

For example, we get two different bases if we pick $\theta = 0$ and $\theta = \pi/4$.

2. If $\vec v = \begin{pmatrix} 3 \\ 2 \\ 1\end{pmatrix}$ and $\vec u = \begin{pmatrix} -1 \\ 0 \\ 1\end{pmatrix}$, find a decomposition

where $\vec v^{||}$ is parallel to $\vec u$ and $\vec v^{\perp}$ is orthogonal to $\vec u$.

Gram-Schmidt says that

Then,

3. For two $m \times n$ matrices $M, N$ we can define the inner product,

Are the vectors,

orthogonal? Explain why or why not.

Yes, because $\langle M, N \rangle = \mathop{tr}(M^T N) = 0$.