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})$$ ^eb068a ### Standard Matrix Representation [Matrix Representation of a Linear Transformation](Matrix%20Representation%20of%20a%20Linear%20Transformation.md) ### Composition Since linear transformations are functions, the concept of [function composition](Function%20Composition.md) also applies: > [!theorem] Composition of Linear Transformations (Functions) ^theorem > >Let $U,V,W$ be [vector spaces](Vector%20Space.md) fixed under the same field, $\mathbb{F}$. Then, let $S: U \to V$ and $T: V \to W$ be two linear transformations: >$$ >S \circ T : U \to W >$$ >$$ >(S \circ T)(\vec{v}) = S(T(\vec{v})) \quad \forall\ \vec{v} \in U >$$ > [!theorem] Composition of Linear Transformations (Matrices) ^theorem > > Let $S: \mathbb{R}^n \to \mathbb{R}^k$ and $T: \mathbb{R}^k \to \mathbb{R}^m$ be two linear transformations in the real set. Then, their **standard matrices** are given by $[S]$ and $[T]$ > $$[S \circ T] = [S][T] >$$ > In other words, the transformation matrix of a composition of transformation is given by [matrix multiplying](Matrix%20Multiplication.md) the two individual transformation matrices. ### Kernel & Image The concepts of [matrix rank](Rank.md) and [nullity](Nullity.md) can be extended to linear transformations as well, albeit in a different sense:   ### Injective & Surjective Because a linear transformation is a function, it can share some similar properties. >[!theorem] Injective Linear Transformations > >A linear transformation is [injective](Function%20-%20Maths.md#injective) if it's [kernel](Kernel.md) only has one element: the zero vector. Let $T: U\to V$ be a linear transformation. Then: >$$ >T \text{ is injective} \iff \text{ker}(T) = \{ \vec{0} \} >$$ >[!theorem] Surjective Linear Transformations > >A linear transformation is [surjective](Function%20-%20Maths.md#Surjective) if it's [image](Image%20-%20Maths.md) is equal to it's codomain. Let $T: U\to V$ be a linear transformation. Then: >$$ >T \text{ is surjective} \iff \text{img}(T) = V >$$ This can give rise to an [Invertible Linear Transformation](Invertible%20Linear%20Transformation.md). ### Examples * [[Differentiation|Differentiation]] * [Integration](Integration)