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## Homework Statement

Let A be a matrix of size 3x3. Let u and v be vectors in R3, such that Au = v. Prove that the matrix equation Ax=2v has a solution.

## Homework Equations

## The Attempt at a Solution

Alright, so this is for an intro linear algebra course, so the solution shouldnt be too coomplicated. But what im confused about, is can i make A equal a matrix, and then solve this problem with actual numbers? Or to prove this, should i be using variables instead???

cause what i did was made:

A =

1 2 3

1 2 3

1 2 3

And then i mad u=

3

2

1

Then i multiplied A by u to get v, and got v=

10

10

10

Then i moved onto proving that Ax=2v has a solution.

So i wrote:

[1 2 3] [x1] = 20

[1 2 3] [x2] = 20

[1 2 3] [x3] = 20

Then i wrote out the matrix:

1 2 3 20

1 2 3 20

1 2 3 20

Passing to RREF i of course got

1 2 3 20

0 0 0 0

0 0 0 0

Which would therefore indicate that with the A and u that i picked, and the other info given that Ax = 2v has a solution because my RREF has 1 basic variable and 2 free variables, with no rows being 0 0 0 ≠ 0.

BUT i just don't know if i should be somehow proving this with variables instead of numbers, cause i see him saying, 'would this be true for a different A and u and v?'

Any help would be appreciated. :) THANKS!