Math V - w5

Linear Spaces

Rⁿ linear space

Cⁿ linear space

When is a subset a linear subspace?

Showing W is a linear space over R

Span of linear space V

Span of linear space V

Examples

Vectors are linearly dependent iif

Vectors are linearly independent iif

Collection of vectors … is a basis of V iif

Dimensions of bases

Dimensions and bases relations

+ finite and infinite dimensional linear space

We consider two subspaces that help us study systems of equations:

- Row spaces

- Column spaces

Row spaces

Row spaces

Row echelon form

A_r denotes REF of A

Column spaces

Column spaces

Row echelon form/pivots

A_r denotes REF of A

Column spaces

Row echelon form/pivots

Example

Relations regarding dimensions

When does system of equations Ax = b have solution?

Null spaces

Fundamental Theorem of Linear Algebra

Example of Finding Null Space

Affine subspaces