5. Linear Transformations

 

Linear Transformations

Definition and Properties

A linear transformation $T: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is a function that preserves vector addition and scalar multiplication:

• $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ for all vectors $\mathbf{u}, \mathbf{v}$ in the domain of $T$.
• $T(c\mathbf{v}) = cT(\mathbf{v})$ for all scalar $c$ and vector $\mathbf{v}$ in the domain of $T$.

Properties of linear transformations include:

• Preservation of zero vector: $T(\mathbf{0}) = \mathbf{0}$.
• Preservation of linear combinations: $T(c_1\mathbf{v}_1 + c_2\mathbf{v}_2) = c_1T(\mathbf{v}_1) + c_2T(\mathbf{v}_2)$.

Kernel and Image of a Linear Transformation

The kernel (null\_space) of a linear transformation $T$, denoted as $\text{ker}(T)$, is the set of all vectors $\mathbf{v}$ in the domain of $T$ such that $T(\mathbf{v}) = \mathbf{0}$.

The image (range) of a linear transformation $T$, denoted as $\text{im}(T)$, is the set of all vectors in the codomain of $T$ that are mapped to by some vector in the domain of $T$, i.e., $\text{im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \text{ is in the domain of } T\}$.

Rank and Nullity, Fundamental Theorem of Linear Algebra

The rank of a linear transformation $T$, denoted as $\text{rank}(T)$, is the dimension of the image of $T$.

The nullity of a linear transformation $T$, denoted as $\text{nullity}(T)$, is the dimension of the kernel of $T$.

The fundamental theorem of linear algebra states that for any linear transformation $T: \mathbb{R}^m \rightarrow \mathbb{R}^n$:

$$ \text{rank}(T) + \text{nullity}(T) = m $$