TITLE: What is the geometric interpretation of the transpose?
QUESTION [194 upvotes]: I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example, if I have a linear transformation, say on the plane, my intuition is to visualize it as some linear distortion of the plane via scaling and rotation. I do not know how this distortion compares to the distortion that results from applying the transpose, or what one can say if the linear transformation is symmetric. Geometrically, why might we expect orthogonal matrices to be combinations of rotations and reflections?

REPLY [12 votes]: Overview If a matrix A acting on vectors tells you how a vector is transformed, the matrix $A^T$ tells you how linear measurement of this vector are transformed.
If E is your vector space, the "space of linear measurements", or dual space is the space of all linear transformation $E \rightarrow \mathbb{R}$. Said differently, it is the space of all linear functions that take in a vector and output a number.
Example Let's work in $\mathbb{R}^2$.

We have a generic vector $X = 
    \begin{pmatrix}
    x_1 \\
    x_2
    \end{pmatrix}$
A sample matrix $A = $$
    \begin{pmatrix}
    1 & 1 \\
    0 & 1
    \end{pmatrix}
$
A sample linear measurement $h(X)=2x_1$, we note it $\begin{pmatrix}
    2 \\
    0
    \end{pmatrix}$

Maybe this measurement has some meaning physically (total mass or momentum...). We want to know how the transformation A affects h.
The familiar product $AX = \begin{pmatrix}
    x_1 + x_2 \\
    x_2
    \end{pmatrix}$ tells you how X is transformed. 
Now that we have seen that X has changed we might ask, what is the new measurement of X by h? It is obtained by computing $h(AX)$:
$h(Ax) = h(\begin{pmatrix}
    x_1 + x_2 \\
    x_2
    \end{pmatrix}) = 2x_1 + 2x_2$ which we note as $\begin{pmatrix}
    2 \\
    2
    \end{pmatrix}$
So when we transform by A, h goes from $\begin{pmatrix}
    2 \\
    0
    \end{pmatrix}$ to $\begin{pmatrix}
    2 \\
    2
    \end{pmatrix}$
It turns out that this transformation is precisely what the transpose is doing, as you can verify: $A^T\begin{pmatrix}
    2 \\
    0
    \end{pmatrix} = \begin{pmatrix}
    2 \\
    2
    \end{pmatrix}$
This is kind of hand-wavy, if you want more details look up dual space, which makes all of this precise.