📂Linear Algebra

Left Multiplication Transformation (Matrix Transformation)

Definition¹

Regarding the field $F$, let’s say $A \in M_{m \times n}(F)$. The following $L_{A}$ is defined as the left-multiplication transformation.

$$\begin{align*} L_{A} : F^{n} &\to F^{m} \\ x &\mapsto Ax \end{align*}$$

Here, $Ax$ is the matrix product of $A$ and $x$.

Description

This is a more abstract description of matrix transformation using the concept of a field.

Theorems

Let’s say $A \in M_{m \times n}(F)$. Then, $L_{A}$ is a linear transformation. Also, when $B \in M_{m \times n}(F)$ and $\beta, \gamma$ are each the standard ordered basis of $F^{n}, F^{m}$, the following holds.

(a) $\begin{bmatrix} L_{A} \end{bmatrix}_{\beta}^{\gamma} = A$

(b) $L_{A} = L_{B} \iff A = B$

(c) If $L_{A + B} = L_{A} + L_{B}$ and $L_{aA} = a L_{A} \forall a \in F$, then

(d) If $T : F^{n} \to F^{m}$ is a linear transformation, there exists a matrix $m \times n$ such that $T = L_{C}$. (In fact, it is $C = \begin{bmatrix} T \end{bmatrix}_{\beta}^{\gamma}$)

(e) If a matrix is referred to as $E$, then $L_{AE} = L_{A} L_{E}$.

(f) If $m=n$, then $L_{I_{n}} = I_{F^{n}}$. Here, the left side’s $I_{n}$ is the $n\times n$ identity matrix, and the right side’s $I_{F^{n}}$ is the $I_{F^{n}} : F^{n} \to F^{n}$ identity transformation.