Homework Help: Linear Algebra Questions

1. Oct 22, 2005

vg19

Hi,
I am having trouble with the following questions
1) Let U be such that AU=0 implies that A=0. If AU=BU, show that A=B.
So far, I did this, but it doesnt seem right to me.
To Show:
A=B
0=A
=AU
=BU
=0
Therefore A = B.
2)If A=
[a b]
[c d] (2x2 matrix, sorry not to sure on how to place them here)

where a is not equal to 0, show that A factors in the form A =
[1 0][y z]
[x 1][0 w] (those are two 2x2 matricies multiplied together)

Im not too sure on how to start on this question at all.

2. Oct 22, 2005

LeonhardEuler

Your mistake in the first one is in assuming A=0. This is not necessarily the case. You want to show that $AU=BU \rightarrow A=B$, or equivalently $AU-BU=0 \rightarrow A=B$. You never said what kind of objects A, B and U are, but presumably they have some sort of linearity property, correct? If this is the case, then it is easy to rearange the equation to get it to say something times U = 0. You are told in the problem that this implies that something is zero. Take it from there.

3. Oct 22, 2005

vg19

Sorry, A, U, and B are matricies. Im not too sure I understand the part when you said A may not be 0. What does it mean when it says "implies that A = 0"?

Thanks!

4. Oct 22, 2005

LeonhardEuler

When they say "AU=0 implies that A=0" they mean that if AU=0, then A=0 where A is any matrix that can be multiplied by U. Don't get confused by the fact that they later use the same letter when they say "If AU=BU, show that A=B". When they say A, they mean any matrix.

5. Oct 23, 2005

HallsofIvy

If AU= BU then AU-BU= 0. That's what you need to use "If AU= 0,...".

6. Oct 23, 2005

vg19

Great! Im pretty sure I understand it now. Thanks a lot