Direct sum of Kernel Transpose and Image

Question

Hi I am having trouble understanding a math assigment.

The assigment is:

If A is a mxn-matrix Show that:

$$ Im(A) ⨁ Ker(A^T) =R^n $$

I have no clue where to start, can anyone push me in the right direction?

Answer 1

Let $x\in\Bbb R^m$, then

$$x\in\ker A^T\iff \forall y\in\Bbb R^n \; \langle A^Tx,y\rangle=0\iff \forall y\in\Bbb R^n \; \langle x,Ay\rangle=0\iff x\in(\operatorname{Im} A)^\perp$$ hence we proved that

$$\ker A^T=(\operatorname{Im} A)^\perp$$ and the result follows if we know that in finite dimensional space and for all subspace $F$ for $\Bbb R^n$:

$$F\oplus F^\perp=\Bbb R^n$$