Linear Vector Spaces

Linear algebra review, part 1: the vector-space vocabulary used throughout the course (part 2 is Matrix Algebra).

Vector Spaces

• A vector space is a collection of objects (“vectors”) that satisfy:
  1. , ,
  3. such that

Real and Complex n-Space

• Typically in this class , for which
• We will sometimes use , for which
• Will use "" to denote vectors with horizontal orientation if needed:

Dot Product

• Dot product:

Subspaces

• is a subspace of , , if satisfies the vector space axioms in its own right
• i.e.

Linear Combinations and Span

• Linear combination: any expression of the form
  • i.e. a weighted sum of vectors
• Span: the span of a collection is the set of all possible linear combinations of the vectors in the collection
  • This defines a subspace of
  • So for example, the above is

Linear Independence and Dimension

• A collection is linearly independent iff
  • i.e. the only possible linear combination with result is the trivial one,
• The dimension of a collection is the maximum number of linearly independent vectors in the collection

Bases

• A collection is a basis for if
  2. The collection is linearly independent
• every can be written as a unique linear combination of the vectors in the basis

NOTE

• A vector space can have many different bases, but all bases will have the same dimension, which defines the dimension of the vector space itself

• For we could use

Possible slip in notes

• The second set is transcribed exactly as written, but those three vectors are linearly dependent (), so they do not actually span . The point stands: any 3 linearly independent vectors form a basis for

Natural Basis

• The first of these is the so-called “natural basis”. Generically, for this is where
• The natural basis is orthonormal:
• Any nonsingular matrix also defines a basis for : see Nonsingular Matrices as a Basis