Transformations

Informally, function is a rule that accepts inputs and produces outputs. For instance, \(f(x)=x^2\) is a function that accepts one number \(x\) as its input, and outputs the square of that number: \(f(2)=4\).

Consider the matrix equation:

\[\mathbf{v} = \mathbf{A}\mathbf{u}\]

If we vary \(\mathbf{u}\), then \(\mathbf{v}\) will also vary; in this way, we think of \(\mathbf{A}\) as function with independent variable \(\mathbf{u}\) (which is a vector in \(\mathbb{R}^n\)) and dependent variable \(\mathbf{v}\) (which is a vector in \(\mathbb{R}^m\)).

At this point it is convenient to fix our ideas and terminology regarding functions, which we will call transformations. This allows us to systematize our discussion of matrices as functions.

u = vec(2, -1, 3)
F = matrix([[2, 1, 0], [0, 1, 0], [0, 0, 0.5]])
u.transform(F)

Vector3(3, -1, 1.5)

Matrices as functions - transformation matrix

If \(\mathbf{A}\) has \(n\) columns, then it only makes sense to multiply \(\mathbf{A}\) by vectors with \(n\) entries. This is why the domain of transformation is \(\mathbb{R}^n\). If \(\mathbf{A}\) has \(m\) rows, then \(\mathbf{A}\mathbf{u}\) has \(m\) entries for any vector \(\mathbf{u}\) in \(\mathbb{R}^n\); this is why the codomain of transformation is \(\mathbb{R}^m\).

Suppose that \(\mathbf{A}\) has columns \(v_1, v_1, \dots v_n\),. If we multiply \(\mathbf{A}\) by a general vector \(\mathbf{u}\), we get:

\[\begin{split}\mathbf{A}\mathbf{u}=\begin{pmatrix}| & | & & | \\ v_1 & v_2 & \dots & v_3\\ | & | & & | \end{pmatrix}\begin{pmatrix}u_1\\ u_2\\ \vdots \\ u_n\end{pmatrix}= u_1v_1 + u_2v_2 + \dots + u_nv_n\end{split}\]

This is just a general linear combination of \(v_1, v_1, \dots v_n\). Therefore, the outputs of matrix transformations are exactly the linear combinations of the columns of transformation matrix \(\mathbf{A}\).

Useful identities

Following identities are often used in linear algebra:

\[\begin{split}\begin{aligned} (\mathbf{A}\mathbf{B})^T &= \mathbf{B}^T\mathbf{A}^T\\ (\mathbf{A}\mathbf{B})^{-1} &= \mathbf{B}^{-1}\mathbf{A}^{-1}\\ (\mathbf{A}^T)^{-1} &= (\mathbf{A}^{-1})^T = \mathbf{A}^{-T}\\ ((\mathbf{A}\mathbf{B})^{-1})^T &= ((\mathbf{A}\mathbf{B})^T)^{-1} = (\mathbf{B}^{-1}\mathbf{A}^{-1})^T = \mathbf{A}^{-T}\mathbf{B}^{-T} \end{aligned}\end{split}\]