Basis

Definition: Basis. A subset of that is: 1) linearly independent, 2) is called a basis of

• Let be a subset of
  • A subset of is a basis of if is linearly independent and
• Examples of basis include , which is a basis of
  • This is called the canonical basis of
  • This is a basis because it is clearly linearly independent, and we can make any vector in from these vectors
• Another example of a basis for includes Vectors and

Proposition: Say we have a matrix , where is the set of all matrices with real numbers (square matrices). If is non-singular, meaning that is square with a non-zero determinant (), the set column vectors and the set of row vectors of are a basis of .

Theorem: Let be a basis of , . Then, any vector can be uniquely (no other form) represented as

• Basis are how we have vector representations in the first place
  • We implicitly use the canonical basis when not specifying basis
• The coordinate with a specific basis is denoted as

Proof: The representation of some is unique. Pick . Since there exists such that . Now to prove uniqueness of the above representation, suppose that an alternate representation . This means that if you subtract the following representations, you get . Since , the basis, is linearly independent, all the coefficients of the vectors must be 0. . Therefore for all thus the representation is unique.

Definition: Dimension of a subspace Let be a linear subspace of . The dimension of S is the number of elements in any basis of . Example: Let . See that is a basis of . So,

• Observations: Any two basis of has the same number of elements
  • Any linearly independent subset of can have at most elements