A transpose of a matrix and its application

Question

Since we know that a matrix can be a tool to describe and change a space,What is exactly a transpose used for and what can we get when we use a transpose to describe a space?Is there a intuitive explanation?

Answer 1

Can be used to find the point in a line closer to a given position outside the line in the plane.

If we consider $f(x)=Ax$ the equation of a line with matrix column $A$ and $b$ is a given point then one seeks for $Ax-b$ to be minimum and that is by finding $x_0$ such that it must satisfy $Ax_0\approx b$, so by multiplication with $A^{\top}$ to get $$A^{\top}Ax_0\approx A^{\top}b$$ this implies $$x_0\approx (A^{\top}A)^{-1}A^{\top}b,$$ that is the instant where $Ax_0$ is closer to $b$.

This is the same to calculate the estimate $x_0\approx\dfrac{A\cdot b}{A\cdot A}$.