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Linear Transformation

Linear Transformations

A Linear Transformation T:V→WT: V \to W is a function between vector spaces that preserves the structure of the space.

Defining Properties

A map TT is linear if:

  1. T(u+v)=T(u)+T(v)T(u + v) = T(u) + T(v)
  2. T(cu)=cT(u)T(cu) = cT(u)

Types of Maps

  • Injective (One-to-One): Kernel is only {0}\{0\}.
  • Surjective (Onto): Image is all of WW.
  • Isomorphism: Both Injective and Surjective.
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Kernel and Image

  • Kernel (Null Space): Everything that gets smashed to zero. Ker(T)={v∣T(v)=0}\text{Ker}(T) = \{ v \mid T(v) = 0 \}.
  • Image (Range): The actual output space. Im(T)={T(v)∣v∈V}\text{Im}(T) = \{ T(v) \mid v \in V \}.

Matrix Representation

Every linear transformation between finite-dimensional spaces can be represented as a matrix .

Basis of V
Transformation Matrix
Basis of W