MATH 2940 Quiz 3

Vector Space Definition

non empty set V of vectors, on which addition and multiplication by a scalar are defined and subject to axioms.

Vector Space Axioms

1. for all vector u and vector v, u + v is also in V.

Properties of Vector Spaces

Subspace Properties

to prove H is a subspace of V, prove all properties and work with generic elements. 

to disprove, show ONE of the properties fails with a concrete counterexample. 

Spans are Subspaces

Proving that a set is a subspace: alternative way

If you can show that a subset H of V is a span of a finite number of vectors in V, then H is a subspace of V.

Null Space

Column Space

The column space of an mxn matrix is a subspace of Rm.

Linear Transformation between vector spaces

Kernel of a Linear Transformation

Range of a Linear Transformation

Row Space

Linear Independence

Basis

is a subset, not a span

standard basis examples

Spanning Set Theorem

Basis of Col(A) Theorem

Basis of Nul(A) Theorem

Row Equivalence effect on Nul(A) and Col(A)

Basis for Row(A) Theorem

Unique Representation Theorem

B coordinates

Order of Basis vectors

Change of Coordinates Matrix

Isomorphism

Coordinate Mapping is an Isomorphism Theorem

number of vectors in V dependence theorem

number of vectors in every basis of V

dimension of a vector space

Dimensions of common vector spaces

Basis theorem

Extending a linearly independent set to a basis theorem

rank-nullity theorem

change of coordinates theorem

properties of chang eof coordinates

invertible matrix theorem continued

Eigenvectors and Eigenvalues

Eigenspace

Eigenvalues of a triangular matrix

eigenvectors corresponding to eigenvalues

characteristic equation

eigenvalues are roots of characteristic polynomial theorem

Multiplicity of Eigenvalue

invertible matrix theorem eigenvalue

matrix similarity

things similar matrices have in common theorem

Diagonalization

Criterion for Diagonalization

Diagonalization Algorithm

Diagonalization Application to Matrix Powers

Does the standard method for finding a spanning set for Nul A always produce a basis?

Row Operation effect on independence/dependence

row ops preserve row space and column dependence (null space), but not the column space itself.

polynomial coordinate mapping example

Plane in R3 isomorphic to R2

Row Rank = Column Rank

yeah

A and AT same/different eigenvalues?