itj5091 – Proofs in the Pudding

Matrices: Surjective, Injective, and Bijective Linear Transformations

June 3, 2023itj5091 1 Comment

IceT Thaewanarumitkul

Transformation is another synonym for “function” but its terminology implies the heavy focus on “movement” of geometric components such as lines, curves, and regions. Today, we will explore the properties of linear transformation which is one of the central topics in linear algebra.

Linear transformations

A linear transformation, in the world of matrices, is a transformation satisfying two conditions.

The transformation preserve the properties of lines and curves.
The origin is fixed.

These two conditions seems complicated to imagine geometrically and we will talk about them numerically instead.

Consider the equation $b = Ax$ when $x$ is a vector in $\mathbb{R}^n$ , $A$ is an $m \times n$ matrix, and $b$ is the output vector in $\mathbb{R}^n$ represents a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ . We also can write the conditions of a linear transformation in terms of function.

Let $T \colon V \to W$ be a linear transformation from $V \in \mathbb{R}^n$ to $W \in \mathbb{R}^m$ . Here, $V$ is called the domain of $T$ and $W$ is called the codomain of $T$ . The range of $T$ , denoted by $\text{range}(T)$ or $\text{Im}(T)$ is the set of all possible outputs, which is a subset of $W$ , defined by

$\text{range}(T)$ = $\{ T(x) \mid x \in V \}$

Since $T$ is a linear transformation, the two conditions of linear transformation still hold and can be described as

$T(kx+ly)$ = $kT(x) + lT(y)$ for all $x, y \in V$ and all $k,l \in \mathbb R$ .
$T(0) = 0$ (The origin).

Looking back to the equation $b = Ax$ , we realize that $b$ is a linear combination of the columns of $A$ ; therefore, $\text{range}(A)$ is the same as the column space of $A$ .… Read the rest