(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A is an c x d matrix. B is a d x k matrix.

If rank(A) = d and AB = 0, show that B = 0.

2. Relevant equations

3. The attempt at a solution

My textbook has a solution but I don't understand it:

The rank of A is d, therefore A is not the zero matrix. (I asked my prof why d can't be equal to zero, he said it just couldn't...?)

If you left multiply A by some elementary matrix to bring it to row echelon form, you get a matrix that looks like:

[ 1 * * * ... *

0 1 * * ... *

0 0 1 * ... *

0 0 0 0 ... 0] (NOTE: * are arbitrary numbers)

And we will write B as a column (1 x k), consisting of [B1, ... , Bd]^{T}

Multiply A and B together, and you get a column that looks like [R1, R2, ... 0, 0, 0]^{T}

For AB = 0, then Ri = 0. Then since A is not zero, B is 0.

This proof seems to make no sense. Why are we writing B as 1 x k? It says in the question B is d x k! Also if A is not zero then why can't you say right off the bat that AB = 0 implies B =0?

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# Homework Help: Linear Algebra Proof (rank)

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