Finding the rank and dimker of a matrix.

Question

Hi so im working on a question about finding the $rank(A)$ and the $dim(Ker(A))$ of a 7x5 Matrix. Without being given an actual matrix to work from.

I have been told that that the homogeneous equation $A\vec x=\vec0$ has general solution $\vec x=\lambda \vec v$ for some non zero $\vec v$ in $R^{5}$.

So my thinking so far is that I know for an $m*n$ matrix we know that:

$rk(A)+dimker(A)=n$ which must mean that $rk(A)+dimker(A)=5$

but this is where I get stuck and dont know how to proceed.

Any help is greately appreciated.

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This is the exact question for the person who asked.

Answer 1

Hint: If you understand the definition of $\ker(A)$ and the definition of dimension, then you can deduce $\dim \ker A$ from the description of the solution to the homogeneous equation.

Answer 2

Is $v$ fixed? If yes, then kernel is generated by only this vector, so 1 dimensional. By the formula you wrote, rank is 4.