Constructing Non-Zero Vectors from a Zero Matrix: A Proof of Linear Dependence

[hkus10]

1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O.
Can you show me the steps of solving this problem?
Please!

 

[Mark44]

1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O.
Can you show me the steps of solving this problem?
Please!

Show us what you've tried. The rules of this forum say that we aren't supposed to provide any help if you haven't given the problem a try.

 

[hkus10]

Show us what you've tried. The rules of this forum say that we aren't supposed to provide any help if you haven't given the problem a try.

Actually, I am stuck how to begin this question. However, I still have some ideas for how to start. I just want to make sure this is the right approach before solving this kind of proof problem. Is it related to the Matrix-Vector Product ?

 

[spamiam]

A vector and an n x 1 matrix are the same thing: it's just semantics.

First, write down what it means for A to be the zero matrix. This will tell you what it is you need to prove. Next, the condition Ax = 0 holds for all n x 1 matrices (or vectors) x, so try to cleverly choose a vector that makes the product Ax nice to work with.

I'm being pretty vague, but I'll be happy to elaborate once you show us your attempt.

 

[hkus10]

Is it I have to converse of the implication first?

 

[spamiam]

I suppose you could, but I think it's easier to prove directly. I guess both methods boil down to the same thing in the end.

Try this: how would you prove that the entry $A_{1,1}=0$? If you can do this, you should be able to generalize your method.

 

[HallsofIvy]

Suppose some entry in A, say A_ij is NOT 0. Can you construct a vector v, such that Av is not 0?