Linear Algebra - Linear Transformations

Linear Transformations

Definition

$T: V \to W$ is linear if:

1. $T(u + v) = T(u) + T(v)$
2. $T(cv) = cT(v)$

Examples:

• $T_A(x) = Ax$
• Rotation, reflection, projection matrices
• Derivative operator

Matrix Representation

For basis $B_V = \{v_1, \dots, v_n\}$ of V and $B_W = \{w_1, \dots, w_m\}$ of W:

Matrix $[T]_{B_W}^{B_V}$: $i$-th column is $[T(v_i)]_{B_W}$

Action: $[T(v)]_{B_W} = [T]\cdot[v]_{B_V}$

Kernel and Image

Kernel (null space): $\ker(T) = \{v \in V : T(v) = 0\}$

Image (range): $\text{im}(T) = \{T(v) : v \in V\}$

Rank: $\dim(\text{im}(T))$

Nullity: $\dim(\ker(T))$

Rank-Nullity Theorem

For $T: V \to W$: $\\dim(\ker(T)) + \dim(\text{im}(T)) = \dim(V)$

One-to-One and Onto

$T$ is one-to-one $\iff \ker(T) = \{0\}$

$T$ is onto $\iff \text{im}(T) = W$

For square matrices: One-to-one $\iff$ onto $\iff$ invertible.

Composition

$(T_2 \circ T_1)(v) = T_2(T_1(v))$

Matrix: $[T_2 \circ T_1] = [T_2][T_1]$