Find a matrix A if given elements of R(A) and Ker(A)

Question

Find a Matrix A ($ Ax = b $) such that:

$u=(1, 1, 5), v=(0, 3, 1) \in R(A) $ and $ w=(1, 1, 2) \in Ker(A) $

I know that if $u, v \in R(A)$ then $Au = b_1$, and $Av = b_2$, and again if $w \in Ker(A)$ so $Aw = 0$, then I could write down a system with many unknowns. Is that the right aprouch ?

Thanks

Answer 1

Guide:

The question just ask you to find one such matrix.

Let $A = \begin{bmatrix} u & v & au+bv \end{bmatrix}$, it has to satisfy $Aw=0$, hence $$u+v+2(au+bv)=0$$

If you can solve for $a$ and $b$, then you have found one such $A$.