Question is simple:
Find a matrix whose right nullspace $\neq$ left nullspace but rowspace $=$ colspace
I thought the inverse symmetric matrix would be a good example for this.
$$
A^T=-A
$$
Let
$$
A=\begin{bmatrix}
0 & 1 & 2 \\
-1 & 0 & +3\\
-2 & -3 & 0 \\
\end{bmatrix}
$$
See that row space = colspace in this case. the basis's are the inverse of each other but they end up forming the same subspace in $\Bbb R^3$
And you can check the nullspace too, left nullspace isn't equal to the nullspace.
But the answer in the textbook says that it is impossible to find such a vector since for colspace to be equal to row space $m=n$
this would make
$m-r=n-r$
this is what the answer says but this is about equality of degrees of the nullspaces right? such a matrix can be found as i gave in an example 🙂
help out pls
Best Answer
Let $\DeclareMathOperator{LNull}{LNull}\LNull(A)$ and $\DeclareMathOperator{RNull}{RNull}\RNull(A)$ denote the left nullspace and right nullspace of $A$ respectively. That is, \begin{align*} \LNull(A) &= \{x : x^\top A=\mathbf 0\} & \RNull(A) &= \{x : Ax=\mathbf 0\} \end{align*}
Your $A$ has $\LNull(A)=\RNull(A)$. Indeed, one may check that $$ \LNull(A)=\RNull(A)=\DeclareMathOperator{Span}{Span}\Span\{\langle3,-2,1\rangle\} $$
In fact, any matrix satisfying $\DeclareMathOperator{Row}{Row}\Row(A)=\DeclareMathOperator{Col}{Col}\Col(A)$ satisfies $\LNull(A)=\RNull(A)$. To check this, recall that for an $n\times n$ matrix $A$ we have \begin{align*} \Row(A) &= \{y^\top A:y\in\Bbb R^n\} & \Col(A) &= \{Ay:y\in\Bbb R^n\} \end{align*} Then note that \begin{align*} x\in\LNull(A) &\Leftrightarrow x^\top A=\mathbf0\\ &\Leftrightarrow x^\top Ay=0&(\forall y\in\Bbb R^n) \\ &\Leftrightarrow \langle x,Ay\rangle=0&(\forall y\in\Bbb R^n) \\ &\Leftrightarrow x\in \Col(A)^\perp \\ &\Leftrightarrow x\in\Row(A)^\perp \\ &\Leftrightarrow \langle x,y^\top A\rangle=0 &(\forall y\in\Bbb R^n) \\ &\Leftrightarrow y^\top Ax=0 &(\forall y\in\Bbb R^n)\\ &\Leftrightarrow Ax=\mathbf0\\ &\Leftrightarrow x\in \RNull(A) \end{align*}