MA1522 3.6 Basis and Coordinates

Basis

Let V be a subspace of Rⁿ. A set S = {u1, ...., uk}, that is a subset of V, is a basis for V if:

1) S spans V (Span(S) = V)

2) S is linearly independent

**If S is a basis for V, every vector in V can be written as a unique combination of vectors in S

*The basis for a subspace may not be unique

Basis for Solution Set of Homogeneous system

The vectors in the general solution for a homogeneous system form a basis for the solution space.

Basis for the zero space {0}

The empty set, ∅.

Bc {0} is linearly dependent. For any set S containing non zero vectors, span(S) is strictly larger than zero space. N

=> No nonempty set can be a basis for the zero space.

-∅ is linearly independent (vacuously)

- Span of S is the smallest subspace V that contains S. The zero space is the smallest subspace containing the empty set, so the span of the empty set is the zero space.

Basis for Rⁿ & Invertibility

A set S = {u1, u2, ...., uk}, is a basis for Rⁿ iff n = k, and the matrix made from its columns (A = (u1 u2 .... uk)) is invertible.

- nxn square matrix A is invertible ←→ its rows/columns are linearly independent

- nxn square matrix A is invertible←→ its rows/columns span Rⁿ

=> nxn square matrix is invertible iff its rows/columns form a basis for Rⁿ

Coordinates relative to a basis

Form augmented matrix with (u1 u2 ... uk| v), rref. [v]s = RHS of rref. Number of coordinates = number of vectors in the basis

*unique iff S is a basis (if not linearly independent, several vectors in R^k may map to the same v∈V.

**order of the vectors in the basis matters for order of coordinates.

"let S = {u1, u2,...., uk} be a basis for a subspace V of Rⁿ. for v∈V, find real numbers c1, c2,...ck s.t. c1u1+c2u2+....+ckuk = v. i.e. solve for (u1 u2 ... uk| v)"