Matrix Transformation

Definition

Let

• $R^n$, $R^m$ : Linear space
• $\mathbf{x}\in R^n$, $\mathbf{w}\in R^m$ : Vectors

If

$$\mathbf{w}=A\mathbf{x}$$

Then

• We view the $\mathbf{w}=A\mathbf{x}$ as the transformation that maps a vector $\mathbf{x}$ in $R^n$ to a vector $\mathbf{w}$ in $R^m$ by multiplying $\mathbf{x}$ on the left by $A$
• We call this a matrix transformation (or matrix operator in case where $m=n$)
• We denote it by $T_A:R^n\to R^n$
• Sometimes we denote it by $\mathbf{x}\stackrel{T_A}{\to }\mathbf{w}$, which is read ”$T_A$ maps $\mathbf{x}$ to $\mathbf{w}$”