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:
1. $T\left(u+v\right)=T\left(u\right)+T\left(v\right)$ for all $u,v\in V$.
2. $T\left(c\cdot v\right)=c\cdot T\left(v\right)$ for all $v\in V$ and scalars $c$.

Properties of Linear Transformations

1. Preservation of Zero Vector: $T\left({\mathbf{0}}_{V}\right)={\mathbf{0}}_{W}$ where ${\mathbf{0}}_{V}$ and ${\mathbf{0}}_{W}$ are zero vectors in $V$ and $W$ respectively.
2. Preservation of Linear Combinations: $T\left({c}_{1}\cdot {v}_{1}+{c}_{2}\cdot {v}_{2}\right)={c}_{1}\cdot T\left({v}_{1}\right)+{c}_{2}\cdot T\left({v}_{2}\right)$ for all ${v}_{1},{v}_{2}\in V$ and scalars ${c}_{1},{c}_{2}$.
3. 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

1. Matrix Transformations: Multiplying a matrix $A$ by a vector $v$ results in a linear transformation of $v$ to a new vector.
2. Rotation and Scaling: Transformations in 2D or 3D space like rotations, scaling, and reflections are often linear transformations.
3. 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 $m×n$ matrix $A$ such that $T\left(\mathbf{v}\right)=A\cdot \mathbf{v}$, 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

1. Rank-Nullity Theorem: For a linear transformation $T:V\to W$, $\text{rank}\left(T\right)+\text{nullity}\left(T\right)=\text{dim}\left(V\right)$, where $\text{rank}\left(T\right)$ is the dimension of the image of $T$ and $\text{nullity}\left(T\right)$ is the dimension of the kernel of $T$.
2. 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 $W$.
3. 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.