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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 $A$

Bernard
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  • 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. – Ben Grossmann Sep 08 '20 at 12:36
  • Relevant posts: [post 1](https://math.stackexchange.com/questions/37398/what-is-the-geometric-interpretation-of-the-transpose), [post 2](https://math.stackexchange.com/questions/2192992/truly-intuitive-geometric-interpretation-for-the-transpose-of-a-square-matrix), [post 3](https://math.stackexchange.com/questions/2782346/what-is-the-intuitive-interpretation-of-the-transpose-compared-to-the-inverse) – Ben Grossmann Sep 08 '20 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). – Ben Grossmann Sep 08 '20 at 12:41
  • It would be very helpful if you could break this question down into a few focused and separately posted questions – Ben Grossmann Sep 08 '20 at 12:43

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