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## Homework Statement

Problem 1:

If A is an

*m*x

*n*matrix and A

**x**=

**0**for all

**x**ε ℝ^n, prove that A = O.

If A and B are

*m*x

*n*matrices and A

**x**= B

**x**for all

**x**ε ℝ^n, prove that A = B.

(O is the 0 matrix,

**x**is the vector x, and

**0**is the 0 vector.)

**2. The attempt at a solution**

First off, I understand the problem intuitively and can make sense of the answer being true. My issue comes in trying to phrase a proof that shows it. I know some sort of explanation of the dot product method of multiplying A by

**x**is necessary, but I can't seem to figure out how to phrase/write/show it. Any help or advice would be much appreciated!

## Homework Statement

Problem 2:

Suppose A is a symmetric matrix satisfying A^2 = O. Prove that A = O. Give an example to show that the hypothesis of symmetry is required.

**2. The attempt at a solution**

Here I know that a symmetric matrix means that A = A^T (its transpose), also that AA = O = (A^T)A, but again I run into the issue of how it is relevant and can be turned into a proof. The problems seemed very straight forward at first glance, so I wasn't motivated to ask my professor about them as I did for others, but when I was confronted with actually writing them out, I didn't know where to begin.

Any help would be wonderful, thank you!