Ch. 4: Vector Spaces

Define: vector space

Define: subspace

Define: null space

Define: column space

Theorem: Is the column space of an m x n matrix a subspace of R^n or R^m?

The column space of an m x n matrix is all of R^m iff...

Define: linear transformation
(think 2 properties)

What makes an indexed set of vectors linearly independent?

Define: basis

The Spanning Set Theorem

A basis for Col A is formed by which columns of matrix A?

Define: The coordinates of a vector x relative to the basis Beta

What kind of transformation is
x -> [x]_B ?

If a vector space V has a basis with n vectors, then any set in V containing more than n vectors must be _____________.

If a vector space V has a basis of n vectors, then every basis of V must contain how many vectors?

Define: Finite Dimensional and Infinite Dimensional vector spaces

Let H be a subspace of V.
What is the relationship between:
dim H and dim V

The Basis Theorem

Define: Rank A

The Rank Theorem:
Rank A + dim Nul A = ?

Invertible Matrix Theorem (continued)

Change of Basis Theorem

(Linear difference equations)
If a_n != 0 and if {z_n} is given...

The set of all solutions to the nth-order homogeneous linear difference equation is: