Linear Transformations

Definition: A transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ (or operator if $T: \mathbb{R}^n \to \mathbb{R}^n$) is defined to be linear if the image $(w_1, w_2, ..., w_m)$ is comprised of only linear equations for every mapping $(x_1, x_2, ..., x_n)$, that is $T(x_1, x_2, ..., x_n) = (w_1, w_2, ..., w_m)$. For any vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$ and any scalar $k$ a transformation is linear if $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$ and $T(k\vec{u}) = kT(\vec{u})$.

Let's first look at an example of a linear transformation. Consider the following linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by the following equations:

(1)

\begin{align} w_1 = x_1 + 3x_2 \\ w_2 = 2x_1 - x_2 \\ w_3 = -x_1 + 4x_2 \end{align}

We note that the equations forming the image, that is $w_1$, $w_2$, and $w_3$ are all linear, so this transformation is also considered linear and that $T(x_1, x_2) = (x_1 + 3x_2, 2x_1 - x_2, -x_1 + 4x_2)$.

For example, if we take the vector $\vec{x} = (1, 2)$ and apply our linear transformation, we obtain a resultant vector $\vec{w} = (7, 0, 7)$, and we say that $(7, 0, 7)$ is the image of $(1, 2)$ under the linear transformation $T$.

In general, a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is generally defined by the following equations:

(2)

\begin{align} \\ w_1 = a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ w_2 = a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \quad \vdots\quad \: \quad \vdots\quad \quad \: \:\vdots\quad \quad \quad \quad \: \: \vdots \quad\\ w_m = a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \\ \end{align}

In matrix notation we can represent this transformation as $w = Ax$. $A$ is called the standard matrix for the linear transformation $T$, though sometimes we use the notation $[ T ]$ instead. Either way, the standard matrix is created from the coefficients from the system of linear equations defining the image of $T$.

(3)

\begin{align} \quad \begin{bmatrix} w_1\\ w_2\\ \vdots\\ w_m \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots\\ x_n \end{bmatrix} \end{align}

This linear transformation $T$ is defined by the standard matrix $A$, so we say that $T$ is multiplication by $A$ and often denote it with the notation $T_A (x) = Ax$. Either way, these transformations will geometrically transform some vector or point in $\mathbb{R}^n$ to some other vector or point in $\mathbb{R}^m$.

Properties of Linear Transformations

We've already stated the following two properties in the definition of a linear transformation, but now we will prove their existence.

Property 1: If $T: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation, then for any vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$ it follows that $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$.

• Proof: Suppose that $T$ is a linear transformation and is multiplication by $A$. Thus it follows that:

(4)

\begin{align} T(\vec{u} + \vec{v}) = A(u + v) \\ T(\vec{u} + \vec{v}) = Au + Av \\ T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v}) \\ \blacksquare \end{align}

Property 2: If $T: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation, then for any vector $\vec{u} \in \mathbb{R}^n$ and any scalar $k$ it follows that $T(k\vec{u}) + kT(\vec{u})$.

• Proof: Suppose that $T$ is a linear transformation and is multiplication by $A$. Thus it follows that:

(5)

\begin{align} T(k\vec{u}) = A(ku) \\ T(k\vec{u}) = k(Au) \\ T(k\vec{u}) = kT(\vec{u}) \\ \blacksquare \end{align}