# Math Help - Euclidean inner product

1. ## Euclidean inner product

Question 1
In each part, use the given inner product on $\mathbb{R}^{2}$ to find ||w||, where w = (-1,3).
(a) the Euclidean inner product
(b) the weighted Euclidean inner product <u,v> = 3u1v1 + 2u2v2, where u = (u1,u2) and v = (v1,v2)
(c) the inner product generated by the matrix $\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$

Question 2
Show that <u,v> = 5u1v1 - u1v2 -u2v1 + 10u2v2 is the inner product on $\mathbb{R}^{2}$ generated by $A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}$

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

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

2. ## Re: Euclidean inner product

instead of using AA, use ATA, so you get a symmetric matrix.