# For each of the following matrices, determine a basis for each of the subspaces R(AT), N(A), R(A), and N(AT): A=begin{bmatrix}3 & 4 6 & 8 end{bmatrix}

For each of the following matrices, determine a basis for each of the subspaces R(AT), N(A), R(A), and N(AT):
$A=\left[\begin{array}{cc}3& 4\\ 6& 8\end{array}\right]$
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nitruraviX

Step 1
Given:
$A=\left[\begin{array}{cc}3& 4\\ 6& 8\end{array}\right]$
Step 2
${A}^{T}=\left[\begin{array}{cc}3& 6\\ 4& 8\end{array}\right]$
Apply ${R}_{2}\to {R}_{2}-\frac{4}{3}{R}_{1}$
$=\left[\begin{array}{cc}3& 6\\ 0& 0\end{array}\right]$
whose rank is 1
$R\left({A}^{T}\right)=\left(\begin{array}{c}3\\ 4\end{array}\right)$
Now, $N\left({A}^{T}\right)$
$3x+6y=0$
$⇒x=-2y$
$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}-2y\\ y\end{array}\right)$

$N\left({A}^{T}\right)=\left(\begin{array}{c}-2\\ 1\end{array}\right)$
$A=\left[\begin{array}{cc}3& 4\\ 6& 8\end{array}\right]$
Apply ${R}_{2}\to {R}_{2}-2{R}_{1}$
$\left[\begin{array}{cc}3& 4\\ 0& 0\end{array}\right]$
Whose rank is 1
$R\left(A\right)=\left(\begin{array}{c}3\\ 6\end{array}\right)$
For N(A)
$3x+4y=0$
$⇒x=-\frac{4}{3}y$
Step 3
$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}-\frac{4}{3}y\\ y\end{array}\right)$

$N\left(A\right)=\left(\begin{array}{c}-\frac{4}{3}\\ 1\end{array}\right)$

Jeffrey Jordon