In each part, use the given inner product on $R^2$ to find $\|\vec w\|$, where $\vec w\ = (-1, 3)$.
Then the problem lists different inner products to use to find the norm but the one I'm having trouble understanding is this:
Use the inner product generated by the matrix A = $\bigl(\begin{matrix}
1&2\\ -1&3
\end{matrix} \bigr)$
I'm sort of at a loss as how to get this into a weighted Euclidean inner product. I know there's a couple equations to use here but I'm not sure how to use them to get what I want, the textbook also points out that the weighted Euclidean inner product is given by a diagonal matrix where everything on the diagonal gives you the weight to use. I think that's what I'm supposed to be looking for, I simply don't understand how to get there from the given matrix.
Best Answer
Could it be that this is the answer?:
$$(x_1,y_1).(x_2,y_2) = (x_1,y_1)\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_2\\ y_2 \end{pmatrix} $$
It is not commutative, but at least it is a single value :-).