For the following matrix, determine a basis for each of the subspace $\displaystyle R(A^{\top})$, $\displaystyle N(A)$, $\displaystyle R(A)$, and $\displaystyle N(A^{\top})$.

$\displaystyle A=\left[\begin {array}{cc}3&4\\\noalign{\medskip}6&8\end{array}\r ight]$

So I found the RREF of A to be $\displaystyle \left[\begin {array}{cc}1&\dfrac{4}{3}\\\noalign{\medskip}0&0\e nd{array}\right]$

$\displaystyle (1,\dfrac{4}{3})$ forms a basis for $\displaystyle R(A)$ and $\displaystyle \left[\begin {array}{c}1\\\noalign{\medskip}\dfrac{4}{3}\end{ar ray}\right]$ forms a basis for $\displaystyle R(A^{\top})$.

$\displaystyle x_{1}+\dfrac{4}{3}x_{2}=0$

Setting $\displaystyle x_{1}=\alpha$ and $\displaystyle x_{2}=\beta$ I got

$\displaystyle \textbf{x}=\alpha\left[\begin {array}{c}0\\\noalign{\medskip}\dfrac{-3}{4}\end{array}\right]+\beta\left[\begin {array}{c}\dfrac{-4}{3}\\\noalign{\medskip}0\end{array}\right]$

Soxis a equal to the N(A)? This is where I am stuck, I don't know where to go from here or even if I begun my process correctly. Help would be greatly appreciated, thank you!