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:

{\begin{aligned}w_{1}=x_{1}+3x_{2}\\w_{2}=2x_{1}-x_{2}\\w_{3}=-x_{1}+4x_{2}\end{aligned}}

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:

{\begin{aligned}\\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{aligned}}

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 $T$ .

{\begin{aligned}\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{aligned}}

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:

{\begin{aligned}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{aligned}}

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 $T(k{\vec {u}})+kT({\vec {u}})$ .