# Kernel and Image

[SOLVED] Kernel and Image

## Homework Statement

Ker(A) = Im(B)
AB = ?

A is an m x p matrix. B is a p x n matrix.

## The Attempt at a Solution

Since Ker(A) is the subset of the domain of B and Im(B) is the subset of the codomain of B, AB = I. I = identity matrix.

Is this right? It doesn't seem to make sense. There must be a mathematical (symbolic) way to solve for AB, right?

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Dick
Homework Helper
I'm assuming this means B:R^n->R^p. A:R^p->R^m. That way at least Ker(A) and Im(B) live in the same space. Now what do you say AB:R^n->R^m is?

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Is AB the p x m identity matrix?

Dick
Homework Helper
Bad guess. AB is nxm. Consider AB(v). B(v) is in Im(B)=Ker(A). Guess again. Uh, what do you mean by 'identity matrix' anyway. There is no pxm identity matrix. Do you mean the zero matrix?

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Since Im(B) = Ker(A), does that imply that A and B are invertible? Then if B(v) is in the Ker(A) and Ker(A) = {0}, then AB = 0. Is that right?

Dick
Homework Helper
Only one thing you said is right. Why should anything be invertible and why should Ker(A)={0}? If B(v) is in Ker(A) what's A(B(v))? If you think AB=0 then try and put together an argument that will convince me. In your own words.

Since B(v) is in Ker(A) and T(v) = A * ker(A) = 0, then AB must be zero.

Dick
Homework Helper
What's T? I think you MIGHT know what you are trying to say, but that's just a guess. If that's what you are planning to turn in as a solution, I don't think it will work. Can't you put that more clearly?

The book defines the kernel as a subspace x and T(x) = Ax = 0 and ker(A) = x. T(x) is the linear transformation and A is the matrix. Should I not put the book's equation T(x) = Ax in my solution?

So in the solution I wrote in my homework, I wrote: BX is in ker(A) and AX = 0, X is in ker(A), therefore AB = 0, is that a good explanation?

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Dick
Homework Helper
I don't have your book. What's the linear transformation associated with Bx? Is it also T? What's the linear transformation associated with AB(x). If it's T, I'll scream. You aren't expressing yourself clearly. That's all I'll say.

Dick
Homework Helper
So in the solution I wrote in my homework, I wrote: BX is in ker(A) and AX = 0, X is in ker(A), therefore AB = 0, is that a good explanation?
It's getting there. Try saying this. For any vector x, Bx is in Im(B). Since Im(B)=Ker(A) then Bx is in Ker(A). If Bx is in Ker(A) then A(Bx)=0. So ABx=0 for all x. So AB=0. How does that sound?

Thank you for your help! I understand what you mean about my vague explanation. I will try to clarify things a bit on my paper.

Dick