Geometrical Representation

Matrix

Matrices are linear transformations on a vector space.

Matrix is basically set of column vectors. And those column vectors basically denotes the bases of coordinate system.

Let $\bf u, \bf v$be the two basis vectors of a coordinate system, then for any point $P$, the coordinates will be written as number of step in the direction of $\bf u$and $\bf v$it will take from origin to point $P$. So,

$$P = x\textbf u+y\textbf v$$

So the cooridnates of point $P$will be $[x,y]$in the coordinate system whose basis vectors are $\bf u, \bf v$

Now let be the matrix containing the bases vector as:

$$M = [\bf u, \bf v] \\ = \begin{bmatrix} u_1 & v_1\\ u_2 & v_2 \end{bmatrix}$$

Now to denote the point P, we show it as matrix multiplication. Essentially $P = M\bf x$where $M$is the basis matrix and $x$is the coordinate vector in the basis $\bf u, \bf v$. So

$$P = M\bf x = [\bf u , \bf v]\begin{bmatrix} x\\ y \end{bmatrix} = x \textbf u+y \textbf v$$

Hence, when it comes to geometry, matrices are merely a notational device of writing down the base vectors of a coordinate system.

Homogeneous coordinates have an additional coordinate marking the coordinate vector as a point (non-zero) or a directional vector (zero). These extended vectors allow translations to be written as matrices as well.

Resources

Matrices from a geometric perspectiveCoranac

A Geometrical Understanding of Matrices

Determinant

Determinant of a matrix basically represents the local, linearized rate of volume change of a transformation.

It's equal to the multiplication of all eigenvalues of the matrix.