- #1

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## Homework Statement

A and B are matrices and x is a position vector. Show that

$$\sum_{v=1}^n A_{\mu v}(\sum_{\alpha = 1}^n B_{v\alpha}x_{\alpha})=\sum_{v=1}^n \sum_{\alpha = 1}^n (A_{\mu v} B_{v\alpha}x_{\alpha})$$

$$= \sum_{\alpha = 1}^n \sum_{v=1}^n(A_{\mu v} B_{v\alpha}x_{\alpha})$$

$$= \sum_{\alpha = 1}^n (\sum_{v=1}^n(A_{\mu v} B_{v\alpha})x_{\alpha})$$

## Homework Equations

N/A

## The Attempt at a Solution

I tried expanding the summations out and multiplying the two brackets each with n terms, but then factorization of the product simply led back to what was first given in the question. I also tried visualizing the problem as matrices, but to no avail. This isn't really homework/part of my portion, but I'm trying to get a hand of tensors and for that I need to understand summations. I'm familiar with basic summation procedures for finding out standard deviation etc but nothing of this sort. Any help is appreciated.