Linear Transformations

Definition: Linear Transformation

A linear transformation from a vector space $(V,F,+,\cdot )$ to a vector space $(W,F,+,\cdot )$ is a function $T:V\to W$ which has the following property for all $\lambda ,\mu \in F$ and all $\mathbf{u},\mathbf{v}\in V$:

$$T(\lambda \mathbf{u}+\mu \mathbf{v})=\lambda T(\mathbf{u})+\mu T(\mathbf{v})$$

Definition: Kernel

Let $(V,F,+,\cdot )$ and $(W,F,+,\cdot )$ be vector spaces.

The kernel of a linear transformation $T:V\to W$ is the set of all vectors $\mathbf{v}\in V$ which the transformation sends to the zero vector in $W$:

$$\{\mathbf{v}\in V\mid T(\mathbf{v})=0_W\}$$

Notation

$$\ker (T)$$

Theorem: Zero Vector to Zero Vector

Every linear transformation $T:V\to W$ always transforms the zero vector $V$ to the zero vector of $W$:

$$T(0_V)=0_W$$

PROOF

TODO

Theorem: Linearity of Composition

If $f:V\to U$ and $g:f(V)\to W$ are linear transformations, then their composition $g\circ f$ is also a linear transformation $g\circ f:V\to W$.

PROOF

$$\begin{array}{rl}g\circ f(\lambda \mathbf{u}+\mu \mathbf{v})& =g(f(\lambda \mathbf{u}+\mu \mathbf{v}))\\ & =g(\lambda f(\mathbf{u})+\mu f(\mathbf{v}))\\ & =g(\lambda f(\mathbf{u}))+g(\mu f(\mathbf{v}))\\ & =\lambda g(f(\mathbf{u}))+\mu g(f(\mathbf{v}))\\ & =\lambda g\circ f(\mathbf{u})+\mu g\circ f(\mathbf{v})\end{array}$$

Theorem: Linearity of Inverse Transformations

If $T$ is a bijective linear transformation, then its inverse $T^{-1}$ is also a bijective linear transformation.

PROOF

TODO

Matrix Representations

Theorem: Matrix Representation of a Linear Transformation

Let $(V,F,+,\cdot )$ and $(W,F,+,\cdot )$ be vector spaces and let $B_V$ and $B_W$ be ordered bases of $V$ and $W$, respectively.

If $T:V\to W$ is a linear transformation, then there exists a matrix $M_T\in F^{\dim (W)\times \dim (V)}$ such that for every $\mathbf{v}\in V$ we have

$${}_{B_W}T(\mathbf{v})=M_T\cdot {}_{B_V}\mathbf{v},$$

where ${}_{B_V}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to $B_V$ and ${}_{B_W}T(\mathbf{v})$ is the coordinate vector of $T(\mathbf{v})$ with respect to $B_W$.

Warning: Dependence on the Choice of Bases

The coefficients of the matrix $M_T$ depend on the choice of $B_V$ and $B_W$, i.e. different ordered bases will make the coefficients of $M_T$ different. To make this clear, we usually denote $M_T$ as ${}_{B_W}[M_T{]}_{B_V}$.

PROOF

TODO

Algorithm: Finding the Matrix Representation

Let $(V,F,+,\cdot )$ and $(W,F,+,\cdot )$ be vector spaces and let $T:V\to W$ be a linear transformation.

We want to find the matrix representation ${}_{B_W}[M_T{]}_{B_V}$ with respect to two ordered bases of our choice: $B_V=({\mathbf{b}}_1^V,\cdots ,{\mathbf{b}}_m^V)$ for $V$ and $B_W=({\mathbf{b}}_1^W,\cdots ,{\mathbf{b}}_n^W)$ for $W$.

1. Determine the effect of $T$ on the elements of $B_V$, i.e. determine ${\mathbf{w}}_1,\dots ,{\mathbf{w}}_m$ by applying $T$ to ${\mathbf{b}}_1^V,\cdots ,{\mathbf{b}}_m^V$.

$${\mathbf{w}}_1=T({\mathbf{b}}_1^V)\cdots {\mathbf{w}}_m=T({\mathbf{b}}_m^V)$$

1. Determine the coordinate vectors ${}_{B_W}{\mathbf{w}}_1,\cdots ,{}_{B_W}{\mathbf{w}}_m$ of ${\mathbf{w}}_1,\cdots ,{\mathbf{w}}_n$ with respect to $B_W$.

• There is no general process for this - it is different for each vector space.

2. Construct ${}_{B_W}[M_T{]}_{B_V}$ by using the coordinate vectors ${}_{B_W}{\mathbf{w}}_1,\cdots ,{}_{B_W}{\mathbf{w}}_m$ as its columns:

$${}_{B_W}[M_T{]}_{B_V}=\left[\begin{array}{ccc}|& |& |\\ {}_{B_W}{\mathbf{w}}_1& \cdots & {}_{B_W}{\mathbf{w}}_m\\ |& |& |\end{array}\right]=\left[\begin{array}{ccc}|& |& |\\ {}_{B_W}T({\mathbf{b}}_1^V)& \cdots & {}_{B_W}T({\mathbf{b}}_m^V)\\ |& |& |\end{array}\right]$$

EXAMPLE

TODO