Linear Transformations
Table of Contents
1. Linear Transformations
We can think of matrices not just as linear combinations, but also as transformations: when a matrix \(A\) multiplies a vector \(\mathbf{v}\), it transforms that vector into another vector \(A\mathbf{v}\). More precisely, a transformation \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\) is a function that takes vectors \(\mathbf{v}\) to vectors \(T(\mathbf{v})\) whose domain is \(\mathbb{R}^n\) and whose codomain is \(\mathbb{R}^m\).
A special type of transformation is a linear transformation, which satisfies the following two conditions:
- \(T(\mathbf{u}+\mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \forall \: \mathbf{u}, \mathbf{v} \in \mathbb{R}^n\)
- \(T(c\mathbf{u}) = c \cdot T(\mathbf{u}) \quad \forall \: c \in \mathbb{R}, \: \mathbf{u}, \mathbf{v} \in \mathbb{R}^n\)
Note that these imply that \(T(0) = 0\), so shift transformations like \(T(\mathbf{v}) = \mathbf{v} + \mathbf{u}_0\) are not linear.
To check if some \(T\) is a linear transformation, we can combine both of the conditions to be one condition:
\begin{align} \boxed{T(c\mathbf{u} + \mathbf{v}) = c \cdot T(\mathbf{u}) + T(\mathbf{v}) \quad \forall \: c \in \mathbb{R}, \: \mathbf{u}, \mathbf{v} \in \mathbb{R}^n} \end{align}1.1. Contraction/Dilation
One common type of linear transformation is a contraction/dilation transformation. This is represented by:
\begin{align} T(\mathbf{u}) = r\mathbf{u} \end{align}where \(r\) is a fixed real number. We can prove that this is a linear transformation by noting:
\begin{aligned} T(c\mathbf{u} + \mathbf{v}) &= r(c\mathbf{u} + \mathbf{v}) \\ &= cr\mathbf{u} + r\mathbf{v} \\ &= c \cdot T(\mathbf{u}) + T(\mathbf{v}) \end{aligned}2. Matrix of a Linear Transformation
Given the standard basis vectors in \(\mathbb{R}^n\), where \(e_i\) has 1 in the i-th position and 0 everywhere else:
\begin{aligned} \mathbf{e_1} &= (1, 0, 0, \dots, 0, 0) \\ \mathbf{e_2} &= (0, 1, 0, \dots, 0, 0) \\ &\vdots \\ \mathbf{e_n} &= (0, 0, 0, \dots, 0, 1) \end{aligned}Notice that we can use these standard vectors to make up any other vector in \(\mathbb{R}^n\); in other words, they span \(\mathbb{R}^n\), as any other vector is a linear combination of these vectors. Therefore, given constants \(c_1, c_2, \dots, c_n\) that make up the vector \(\mathbf{v}\) and linear transformation \(T\):
\begin{aligned} T(\mathbf{v}) &= T(c_1\mathbf{e_1} + c_2\mathbf{e_2} + \cdots + c_n\mathbf{e_n}) \\ &= c_1T(\mathbf{e_1}) + c_2T(\mathbf{e_2}) + \cdots + c_nT(\mathbf{e_n}) \end{aligned}This is basically a linear combination of the vectors \(T(\mathbf{e_1}), T(\mathbf{e_2}), \dots, T(\mathbf{e_n})\). Thus, we can rewrite any linear combination with a matrix \(A\) made up of these vectors:
\begin{align} T(\mathbf{x}) = A\mathbf{x} \end{align}where \(A = \begin{bmatrix} T(\mathbf{e_1}) & T(\mathbf{e_2}) & \cdots & T(\mathbf{e_n}) \end{bmatrix}\).
As such, this gives us a powerful way to look at linear transformations: they are simply a scaling or combination of the basis vectors.
2.1. Existence and Uniqueness
The concept of a linear transformation allows us to understand existence and uniqueness through a different perspective.
We say a function \(f: A \rightarrow B\) is onto (surjective) if the codomain \(B\) is the same as the range; i.e. every point in the codomain is achieved by this function. Equivalently, a transformation \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\) is onto \(\mathbb{R}^m\) if each \(\mathbf{b}\) in \(\mathbb{R}^m\) is the image of at least one \(\mathbf{x}\) in \(\mathbb{R}^n\). This is a question of existence: the mapping \(T\) is not onto if there is some \(\mathbf{b}\) in \(\mathbb{R}^m\) such that the equation \(T(\mathbf{x}) = \mathbf{b}\) has no solution.
We can also view this question as, do the columns of \(A\) (the equivalent matrix for \(T\)) span \(\mathbb{R}^m\)? In other words, a linear transformation is onto if there a pivot in every row of the matrix \(A\).
We say a function \(f: A \rightarrow B\) is said to be one-to-one (injective) if any two inputs give different outputs, i.e. \(a_1 \neq a_2 \Rightarrow f(a_1) \neq f(a_2)\). Equivalently, a transformation \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\) is one-to-one if each \(\mathbf{b}\) in \(\mathbb{R}^m\) is the image of at most one \(\mathbf{x}\) in \(\mathbb{R}^n\). This is a question of uniqueness: for each \(\mathbf{b}\) in \(\mathbb{R}^m\), the equation \(T(\mathbf{x}) = \mathbf{b}\) has either one solution, or no solution at all.
To check if a linear transformation is one-to-one, we can check to see if \(T(\mathbf{x_1}) = T(\mathbf{x_2}) \Rightarrow \mathbf{x_1} = \mathbf{x_2}\). This is equivalent to asking whether the equation \(T(\mathbf{x_1} - \mathbf{x_2}) = 0\) has only one solution: the trivial solution. In other words, a linear transformation is one-to-one if there is a pivot in every column of the corresponding matrix \(A\).