When A moves from one side of the dot product to the other, it becomes $A^T$ I'm going through the book Gilbert Strang - Introduction to Linear Algebra, Third Edition (2003) and in the section for transpose matrices right near the end, there is this paragraph: ![paragraph]( I completely understand the transformation, but I don't understand what the 'deep purpose' here really is. Also, I don't understand in the last sentence how "$A$ moves from one side of the dot product to the other". What i see in $(Ax)^Ty$ is: $A$ firstly linearly transforms the vector **x** and then the resulting vector is turned into a linear transformation on its own (thanks to the transpose operation). Now, **y** is linearly transformed with this new transformation and the end result is a single number. My point is, there was never explicit dotting between **x** and **y** in the first place, so what does the author mean $A$ going from one side to the other?

By dotting, the author probably means that $\vec{a} .\vec{b} = a^Tb$, for _vectors_ $a$ and $b$, so

$(Ax)^Ty = (Ax).y$,

and $x^T(A^T y ) = x.(A^Ty)$, note that $Ax$ and $A^Ty$ are vectors.