This is Gilbert Strang's four subspaces of a matrix
Is there a way to plug in the concrete example into the visualization for these four subspaces?
for instance, to svd this matrix $$ \begin{bmatrix} 1 & 1 \\ 2 & 2 \\ 0 & 0 \end{bmatrix} $$
we use the eigendecomposition of this matrix $$\begin{bmatrix}1 & 2 & 0\\1 & 2 & 0\end{bmatrix} \begin{bmatrix} 1 & 1 \\ 2 & 2 \\ 0 & 0 \end{bmatrix} = \displaystyle \left[\begin{matrix}5 & 5\\5 & 5\end{matrix}\right]$$
In this process, is there a way to plug in the matrix $\displaystyle \left[\begin{matrix}5 & 5\\5 & 5\end{matrix}\right]$ into the visualization for these four subspaces?
one valid SVD for this matrix would be
$u_2, u_3$ are one of orthonormal bases of the null space of $A^T$.
is there a way to visualize this null space of $A^T$?
Suppose that $A$ is $m \times n$ with rank $r$ (i.e. $r$ is the number of non-zero singular values), and suppose $A = U \Sigma V^T$ is a singular value decomposition.
In this case, the first $r$ columns of $U$ form a basis (an orthonormal basis) of the column-space, and the remaining $m-r$ columns form a basis of the null-space of $A^T$. The first $r$ columns of $V$ form a basis of the row-space, and the remaining $n-r$ columns of $V$ form a basis of the null-space of $A$.
I'm not quite sure if this helps you "visualize" anything, but perhaps this is the kind of thing you're looking for.