# The underlying significance of a Transpose Matrix

I have been re-studying Linear Algebra and have been curious about the real significance of the Transpose Matrix. It is easy to do, but there must be a reason for doing it right?

If a transformation vector $$X$$ is not multipliable with transformation Matrix A, what is the geometric, realistic meaning of $$A^T X$$? I mean in terms of spans, basis, column space and utility. So for the sake of argument, say A does some sheering. What would $$A^T$$ be doing to $$X$$?

The second question was with respect to left Null spaces, I can see how say the left null space is perpendicular to the column space. So that might give a good complete estimate of the geometry of the range.

And similarly, I am guessing the Row space and Null spaces do the same for the domain.

My real question is how did all this come up? I feel like there is a little gap or a hole in how I understand these things. Maybe an explanation of how someone thought to actually define row spaces and how it backtracked to being related to a transpose matrix.

Another interesting definition I am curious about is how if A isn't even invertible, but its columns are independant, then $$A^TA$$ happens to be invertable. What does that mean to the range of $$A^TA$$ And how is it different from the Range of $$A^T$$ or the Domain of $$A$$

• If $A$ represents a shear, then it is a square matrix. So, for a column vector $X$, $AX$ is defined if and only if $A^TX$ is defined. Commented Sep 8, 2020 at 12:36
• Relevant posts: post 1, post 2, post 3 Commented Sep 8, 2020 at 12:40
• For your last question: in all cases, $A^TA$ has the same row-space and nullspace as $A$ (but not necessarily the same column-space or left nullspace). Similarly, $AA^T$ has the same column space and left nullspace as $A$ (but not necessarily the same row-space or nullspace). Commented Sep 8, 2020 at 12:41
• It would be very helpful if you could break this question down into a few focused and separately posted questions Commented Sep 8, 2020 at 12:43