Question 1

In each part, use the given inner product on $\displaystyle \mathbb{R}^{2}$ to find ||w||, where w = (-1,3).

(a) the Euclidean inner product

(b) the weighted Euclidean inner product <u,v> = 3u_{1}v_{1}+ 2u_{2}v_{2}, where u = (u_{1},u_{2}) and v = (v_{1},v_{2})

(c) the inner product generated by the matrix $\displaystyle \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$

Question 2

Show that <u,v> = 5u_{1}v_{1}- u_{1}v_{2}-u_{2}v_{1}+ 10u_{2}v_{2}is the inner product on$\displaystyle \mathbb{R}^{2}$ generated by$\displaystyle A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}$

Attempt:

$\displaystyle \begin{bmatrix} v_{1} & v_{2} \end{bmatrix} \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}\begin{bmatrix} u_{1} \\u_{2} \end{bmatrix}$

but i couldn't get the equation above. I suppose that the above formula only hold for diagonal matrix?