In this section:

  • and are vector spaces over
  • and
  • and are bases of and . (resp.)

Linearity

  • Given a function . The following statements are equivalent:

    • is a linear transformation
    • (9.1.1) is additive and homogeneous
      • Additivity:
      • Homogeneity:
    • (9.1.3)
  • Theorems

    • (9.1.2a) If , then is not linear
    • (9.7.1) let and linear transformations, then is also linear transformation

Hom

  • is the set of all linear transformations from to
  • (9.7.6) is a vector space
    • and are isomorphic vector spaces
    • Vector space operations: (see q9.7.2-3)
      • Addition:
        • Commutativity:
        • Associativity:
        • Identity:
        • Inverse:
      • Scalar multiplication:
        • Distributivity (vector (transformations) addition):
        • Distributivity (field addition):
        • Compatibility:
        • Identity:
    • Vector space properties of
      • or

Composition

Equality

  • (9.4.1) , and spans . then

Transformation matrix

  • (d10.1.1) The transformation matrix of the linear transformation , relative to the bases and , is the matrix (with entries in ) defined by
  • is the transformation matrix of by the bases and if and only if

Linear Transformation

In this section:

Fundamental Spaces

Image

  • is column-space of

  • (9.3.6) let , and , then,

  • (9.5.6) if is injective, and linearly indep., then is also linearly indep.

Kernel

  • is equal to the null space of

Rank

Surjective (Onto)

The following statements are equivalent:

  • is surjective (epimorphism)
  • columns spans
  • (T is right-cancellable)
  • (T is right-invertible) There exists such that
  • has full row rank

Theorems:

  • if then cannot be onto

Injective (One-to-One)

The following statements are equivalent:

  • is injective (monomorphism)
  • (9.5.2)
  • The colmuns of are linearly independent
  • (T is left-cancellable)
  • (T is left-invertible) There exists such that
  • has full column rank

Theorems: ( is injective)

  • if then cannot be one-to-one
  • (q9.6.3c) are linearly independent, if and only if,

Isomorphism

dimV=dimW

In this section, (), that is is a square matrix.

Theorems

  • (9.4.2) Let be a basis of and an arbitrary list of vectors in . Then there exists a unique linear map such that

Isomorphism

  • (9.6.2) The following statements are equivalent
    • is a linear isomorphism (bijective linear transformation) from on
    • is injective
    • is surjective
    • is bijective
    • There is such that
    • There is such that
    • (10.5.1-2) is invertible, i.e. exists
    • (9.9.2) the inverse exists, such that and

Linear Endomorphism

In the case where , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term linear operator can have different meanings for different conventions (wikipedia)

Eigenvalues

Equivalent definitions of eigenvalue for the linear transformation .

  • (d11.2.1) is an eigenvalue of
  • (d11.2.1) There exists a non-zero vector in such that .
  • (In such a case, is called an eigenvector of related to the eigenvalue )
  • The operator is singular.
  • has nontrivial solutions, i.e.,
  • (11.4.1) The characteristic equation
  • is a root of the characteristic equation

Theorems:

  • (q11.2.4a) if is an eigenvalue of , then for each , is an eigenvalue of
  • (q11.2.4b) if is an eigenvalue of , then , is a eigenvalue of . (for each natural )
  • (11.2.6) has at most distinct eigenvalues
  • if for some natural , then has at most the eigenvalues (todo by q11.2.4)
Eigenvectors

Definitions of eigenvector. The following statements are equivalent:

  • (d11.2.1) is an eigenvector of related to .
  • (d11.2.1) is a non-zero vector in such that .
Eigenspace

Definitions of the eigenspace of associated with its eigenvalue .

  • (d11.2.2)

Diagonalizability

  • (d11.1.1) is diagonalizable

  • has a basis such that the matrix of (by that basis), is diagonal

  • (11.2.3) has a basis in which all its vectors are eigenvectors of

  • (11.2.5) if has distinct eigenvalues, then is diagonalizable #todo https://textbooks.math.gatech.edu/ila/diagonalization.html

characteristic polynomial

  • (d11.4.4) the characteristic polynomial of is the characteristic polynomial of the transformation matrix by some basis
  • 2