Linear Transformation

A linear transformation is any type of function that operates over a vector space and preserves it. In other words, a transformed vector space is still a vector space and still follows the axioms of a vector space.

See also Geometric Transformation

Definition §

Let $V_{\mathbb{F}}$ and $W_{\mathbb{F}}$ be two vector spaces fixed over the field, $\mathbb{F}$. A linear transformation, $T$, from $V_{\mathbb{F}}$ to $W_{\mathbb{F}}$ is notated as:

$$T:V_{\mathbb{F}}\to W_{\mathbb{F}}$$

And it must preserve vector addition and scalar multiplication. If $\vec{u},\vec{v}\in V_{\mathbb{F}}$ and $k\in \mathbb{F}$:

$$T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$$

$$T(k\vec{v})=k\cdot T(\vec{v})$$

More tersely, we can say a linear transformation is one that preserves linear combinations:

$$T(k_1\vec{u}+k_2\vec{v}+\dots )=k_{1}T(\vec{u})+k_{2}T(\vec{v})+\dots$$

Furthermore, the zero vector of $V$ is always mapped to the zero vector of $W$: $T({\vec{0}}_V)=T({\vec{0}}_W)$

Standard Matrix Representation §

Matrix Representation of a Linear Transformation

Composition §

Since linear transformations are functions, the concept of function composition also applies:

Composition of Linear Transformations (Functions) ^theorem

Let $U,V,W$ be vector spaces fixed under the same field, $\mathbb{F}$. Then, let $S:U\to V$ and $T:V\to W$ be two linear transformations:

$$S\circ T:U\to W$$

$$(S\circ T)(\vec{v})=S(T(\vec{v}))\forall \vec{v}\in U$$

Composition of Linear Transformations (Matrices) ^theorem

Let $S:{\mathbb{R}}^n\to {\mathbb{R}}^k$ and $T:{\mathbb{R}}^k\to {\mathbb{R}}^m$ be two linear transformations in the real set. Then, their standard matrices are given by $[S]$ and $[T]$

In other words, the transformation matrix of a composition of transformation is given by matrix multiplying the two individual transformation matrices.

Kernel & Image §

The concepts of matrix rank and nullity can be extended to linear transformations as well, albeit in a different sense:

Injective & Surjective §

Because a linear transformation is a function, it can share some similar properties.

Injective Linear Transformations

A linear transformation is injective if it’s kernel only has one element: the zero vector. Let $T:U\to V$ be a linear transformation. Then:

$$T\text{ is injective}⟺\text{ker}(T)=\{\vec{0}\}$$

Surjective Linear Transformations

A linear transformation is surjective if it’s image is equal to it’s codomain. Let $T:U\to V$ be a linear transformation. Then:

$$T\text{ is surjective}⟺\text{img}(T)=V$$

This can give rise to an Invertible Linear Transformation.

Examples §

• Differentiation
• Integration