Linear Transformations

Definition

  • Linear Transformation: A linear transformation T:VW between vector spaces V and W preserves vector addition and scalar multiplication, satisfying two key properties:
    1. T(u+v)=T(u)+T(v) for all u,vV.
    2. T(cv)=cT(v) for all vV and scalars c.

Properties of Linear Transformations

  1. Preservation of Zero Vector: T(0V)=0W where 0V and 0W are zero vectors in V and W respectively.
  2. Preservation of Linear Combinations: T(c1v1+c2v2)=c1T(v1)+c2T(v2) for all v1,v2V and scalars c1,c2.
  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:RnRm can be represented by an m×n matrix A such that T(v)=Av, where v is a vector in Rn.
  • The columns of the matrix A are the images of the standard basis vectors in Rn under the transformation.

Fundamental Theorems

  1. Rank-Nullity Theorem: For a linear transformation T:VW, rank(T)+nullity(T)=dim(V), where rank(T) is the dimension of the image of T and nullity(T) 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.