Suppose a certain matrix, $A$, contains vectors $v_n$, with $x$ number of elements. Let's say there are two independent vectors ($v_1$ and $v_2$) with $2$ elements each. Visually this forms a plane in $R^2$.
Now let's say we have a matrix $B$, a $2$ x $3$ matrix. Now how would I go about visualizing $B$? Can I just take the transpose of $B$ ($B^T$) and make it a $3$ x $2$ and visualize it as a subspace in $R^3$?
Just scratching linear algebra here. These are thoughts and ideas I had questions about.
It depends on what do you want to visualize.
If you want to visualize some elements of the column space, then each column is a vector in $\mathbb{R}^2$, and you can plot $3$ vectors there.
If you are interested in visualizing some elements of the row space, then each row is a vector in $\mathbb{R}^3$, and you have two vectors in $\mathbb{R}^3$.