# 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}1 & 3&1 2 & 4&0end{bmatrix}

For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A), and N(AT ):
$A=\left[\begin{array}{ccc}1& 3& 1\\ 2& 4& 0\end{array}\right]$
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Laaibah Pitt
Step 1
Given:
$A=\left[\begin{array}{ccc}1& 3& 1\\ 2& 4& 0\end{array}\right]$
The reduced row echelon form of $A=\left[\begin{array}{ccc}1& 3& 1\\ 2& 4& 0\end{array}\right]$ is $\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 1\end{array}\right]$
Since (-1,0,-2) and (0,1,1) form a basis for the row space of matrix A , we have $\left\{\left(1,0,-2{\right)}^{T},\left(0,1,1{\right)}^{T}\right\}$ form a basis for $R\left({A}^{T}\right)$
When from the reduced row echelon form of matrix A , we have,
${x}_{1}-2{x}_{3}=0$
$⇒{x}_{1}=2{x}_{3}$
${x}_{1}+{x}_{3}=0$
$⇒{x}_{1}=-{x}_{3}$
Step 2
Set ${x}_{3}=\alpha$. Then N(A) consists of all vectors of the form $\alpha \left(2,-1,1{\right)}^{T}$
Therefore, $\left(2,-1,1{\right)}^{T}$ is its basis
Now ${A}^{T}=\left[\begin{array}{cc}1& 2\\ 3& 4\\ 1& 0\end{array}\right]$
The reduced row echelon form of
Since (1,0) and (0,1) form the basis for the row space of matrix ${A}^{T}$ , we have $\left\{\left(1,0{\right)}^{T},\left(0,1{\right)}^{T}\right\}$ form a basis for R(A).
When $x\in N\left({A}^{T}\right)$ from the reduced row echelon form of matrix ${A}^{T}$ , we have,

Step 3 It follows that $N\left({A}^{T}\right)=0$
Therefore , there is no basis for $N\left({A}^{T}\right)$
Jeffrey Jordon