MATH 323 Lecture 10

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Vector Space

Closure criteria:


A vector space automatically has two subspaces:

When and , then is called a proper subspace.

Null Space of Matrix

Let be a matrix, and let be the set of all solutions to the system :

is clalled the null space (also called nullspace or kernel) of .

is a vector space since and


Determine if

Gauss-Jordan reduction of gives , so the solutions of are

Linear Combinations and Span

Let .

The sum , where , is called a linear combination of .

The set of all such linear combinations is called the span of :


and are in

The span of and is the set of all vectors .

These two vectors form the standard basis in 2D space.


is a subspace of

Spanning Set

The set is a spanning set for if

If a subset of is a spanning set of , then itself is a spanning set of (set linear coefficient of other terms in set to 0).


These are the standard basis vectors for 3D space.


Is a spanning set of ?

This has a solution for any , so it is indeed a spanning set (the vectors in the original problem are noncoplanar, so that's easier to visualize)


is the set of polynomials of degree < 3.

Linear Independence

Given , , and ,

This is true since

The set is called linearly dependent

  1. If span and one of these vectors can be written as a linear combination of the others, then those vectors span .
  2. Given vectors , it is possible to write one of the vectors as a linear combination of the other vectors iff there exist scalars (not all zero!) such that .