Linear Transformations

Linear Transformation

Definition:
A mapping L from a vector space V into a vector space W is said to be a linear transformation if

for all and for all scalars and .

  • Alternative definition:
    L is a linear transformation if and only if for all and scalars 𝛼,

Mapping

  • Definition:
    A mapping L from a vector space V into a vector space W will be denoted
    When the arrow notation is used, it will be assumed that V and V represent vector spaces.

linear operator

  • Definition:
    When the vector spaces V and W are the same, we will refer to a linear transformation as a linear operator on V. Thus, a linear operator is a linear transformation that maps a vector space V into itself.

L𝑖n𝑒𝑎𝑟 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 ⊂ 𝑙𝑖n𝑒𝑎𝑟 𝑡𝑟𝑎n𝑠𝑓𝑜𝑟m𝑎𝑡𝑖𝑜n

Linear Transformations from to

  • Definition:
    If A is any m×n matrix, and define
    via
    for all . Then L is a linear transformation from to .
  • Property:
    If L is a linear transformation mapping a vector space V into a vector space W, then
    • (where and are the zero vectors in V and W, respectively).
    • If are elements of V and are scalars, then =
    • .

Kernel

  • Definition:
    Let L : be a linear transformation. The kernel of L, denoted ker⁡(L), is defined by
    • = {}
  • The kernel of L is the set of all vectors from V that L transforms to the zero vector in W.
  • Note, if L is defined via matrix multiplication for some matrix A, then ker⁡(L)=𝑁(A), the null space of A.

Image

  • Definition:
    Let L : V \rightarrow W be a linear transformation and let S be a subspace of V. The image of S, denoted L(S), is defined by
    = { for some $ v\in S$}

Range

  • Definition:
    The image of the entire vector space, L(V), is called the range of L.

In other word

  • If {} is a spanning set of S, then L(S) is the span of .
  • If L is defined via matrix multiplication for some matrix A, then L(V), the range of L, is the column space of A.

Theorem 2.1.1 (Kernel/Image are subspaces)

If L : V \rightarrow W is a linear transformation and S is a subspace of V. Then

  • ={} is a subspace of V.
  • ={ for some } is a subspace of W.

Matrix Representations of Linear Transformations

Theorem 2.2.1 (Standard matrix representation)

If L is a linear transformation mapping into , then there is an m×n matrix A, such that for each , L(x)=Ax.
In fact, the column vector of A is given by (j=1,2,…,n)

  • How to write the m×n matrix A:
    • See what L does to the basis vectors in V. Because is in W, write it as linear combination of ;
    • write the coordinate vector as the first column of matrix A.
    • Repeat for , to find the ,… column of A.

Theorem 2.1.2

If 𝐸={} and 𝐹={} be ordered bases for and , respectively. If L: is a linear transformation and A is the matrix representing L w.r.t. 𝐸 and 𝐹, then
(j=1,2,…,n)
Where W=.

Corollary

If A is the matrix representing the linear transformation L: with respect to the bases
𝐸={} and 𝐹={}
then the reduced row echelon form of [|] is .

Table Of Contents
Linear Transformations
Linear Transformation
Mapping
linear operator
Linear Transformations from to
Kernel
Image
Range
Theorem 2.1.1 (Kernel/Image are subspaces)
Matrix Representations of Linear Transformations
Theorem 2.2.1 (Standard matrix representation)
Theorem 2.1.2
Corollary