What does $x^TA$ mean intuitively?

Question

$Ax$ means apply transformation $A$ on $x$, we can look at the null space and range of A and have an understanding of what's going to happen to the vector after we apply transformation $A$.

Just wondering what the equivalent reasoning is for $x^TA$?

I am asking because in my computational optimization course we are often asked to prove properties of the quadratic form $x^TAx$ which involves computing $x^TA$. Apologies if I am missing something and this is trivial

Answer 1

Just as $Ax$ takes a linear combination of the columns of $A$ (summing $x_j$ times the $j$th column), $x^\top A$ takes a linear combination of the rows of $A$ (summing $x_i$ times the $i$th row).

Nevertheless, I think the right answer to your question about the quadratic form $x^\top Ax$ is to interpret $$x^\top Ax = x\cdot Ax$$ as the dot product of two vectors in $\Bbb R^n$, giving you the quadratic function $\sum\limits_i x_i\sum\limits_j a_{ij}x_j = \sum\limits_{i,j}a_{ij}x_ix_j$.

Answer 2

$x^TA$ is a transformation $A$ on $x^T$, with column operations instead of row operations.

Answer 3

$X^T$ is defined as the transpose of $X$, so if $X$ is a vertical vector, it would become horizontal starting with the value at the top of the vector and vice versa if it's horizontal. It's mainly so that you are able to do vector-matrix multiplication and in order to do vector-matrix multiplication, the number of columns in the matrix must correspond to the number of rows in the vector.