# Linear Transformations

#### Definition

**Linear Transformation:** A linear transformation $T:V\to W$ between vector spaces $V$ and $W$ preserves vector addition and scalar multiplication, satisfying two key properties:
- $T(u+v)=T(u)+T(v)$ for all $u,v\in V$.
- $T(c\cdot v)=c\cdot T(v)$ for all $v\in V$ and scalars $c$.

#### Properties of Linear Transformations

**Preservation of Zero Vector:** $T({\mathbf{0}}_{V})={\mathbf{0}}_{W}$ where ${\mathbf{0}}_{V}$ and ${\mathbf{0}}_{W}$ are zero vectors in $V$ and $W$ respectively.
**Preservation of Linear Combinations:** $T({c}_{1}\cdot {v}_{1}+{c}_{2}\cdot {v}_{2})={c}_{1}\cdot T({v}_{1})+{c}_{2}\cdot T({v}_{2})$ for all ${v}_{1},{v}_{2}\in V$ and scalars ${c}_{1},{c}_{2}$.
**Image and Kernel:** The image of $T$ is the set of all vectors in $W$ that are the output of $T$ applied to some vectors in $V$. The kernel of $T$ is the set of all vectors in $V$ mapped to the zero vector in $W$.

#### Examples of Linear Transformations

**Matrix Transformations:** Multiplying a matrix $A$ by a vector $v$ results in a linear transformation of $v$ to a new vector.
**Rotation and Scaling:** Transformations in 2D or 3D space like rotations, scaling, and reflections are often linear transformations.
**Projection:** Mapping vectors onto a subspace (e.g., projecting onto a line or a plane) is a linear transformation.

#### Matrix Representation of Linear Transformations

- A linear transformation $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ can be represented by an $m\times n$ matrix $A$ such that $T(\mathbf{v})=A\cdot \mathbf{v}$, where $\mathbf{v}$ is a vector in ${\mathbb{R}}^{n}$.
- The columns of the matrix $A$ are the images of the standard basis vectors in ${\mathbb{R}}^{n}$ under the transformation.

#### Fundamental Theorems

**Rank-Nullity Theorem:** For a linear transformation $T:V\to W$, $\text{rank}(T)+\text{nullity}(T)=\text{dim}(V)$, where $\text{rank}(T)$ is the dimension of the image of $T$ and $\text{nullity}(T)$ is the dimension of the kernel of $T$.
**Injective and Surjective Transformations:** A linear transformation is injective (or one-to-one) if its kernel contains only the zero vector; it is surjective (or onto) if its image spans the entire codomain $W$.
**Isomorphism:** A bijective linear transformation between vector spaces, where both the transformation and its inverse are linear, is called an isomorphism. Isomorphisms preserve both the structure and dimensionality of vector spaces.