What does the transpose of a matrix represent?

Question

What is the significant of the transpose of a linear transformation's matrix? If the matrix is equal to its transpose, its symmetric, but does that have any importance in what the transformation does?

Answer 1

A quick answer is that if $A$ is a real $m \times n$ matrix, then the key property of $A^T$ is that $$ \langle A^T z, x \rangle = \langle z, Ax \rangle $$ for all $x \in \mathbb R^n, z \in \mathbb R^m$. Often this is the best way to think about $A^T$.

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Here's a more abstract viewpoint:

Let $V$ and $W$ be vector spaces over a field $F$, and let $T:V \to W$ be a linear transformation. Then if $z$ is a linear functional on $W$, we can define a linear functional $T^*z$ on $V$ as follows: $$ \tag{$\spadesuit$}T^*z(x) = z(Tx) \qquad \text{for all } x \in V. $$ The function $T^*:W^* \to V^*$ is a linear transformation, which could be called the "dual" of $T$ or the "transpose" of $T$. If we use the suggestive notation $$\langle z, w \rangle := z(w) $$ then equation ($\spadesuit$) becomes $$ \langle T^*z, x \rangle = \langle z, Tx \rangle \qquad \text{for all } x \in V. $$

Now suppose that $U$ and $V$ are finite dimensional and $\alpha$ and $\beta$ are ordered bases of $U$ and $V$, respectively. Let $\alpha^*$ and $\beta^*$ be the corresponding dual bases. If the matrix representation of $T$ with respect to $\alpha$ and $\beta$ is $A$, then you can check that the matrix representation of $T^*$ with respect to $\beta^*$ and $\alpha^*$ is $A^T$, the transpose of $A$.

The mapping $T^*$ is what's really fundamental. $A^T$ is just a matrix that represents $T^*$.