4 Vector Spaces 4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transformations 4.3 Linearly Independent Sets; Bases 4.4 Coordinate systems

Definition Let H be a subspace of a vector space V. An indexed set of vectors in V is a basis for H if i) is a linearly independent set, and ii) the subspace spanned by coincides with H ; i.e. REVIEW

The Spanning Set Theorem Let be a set in V, and let. a.If one of the vectors in S, say, is a linear combination of the remaining vectors in S, then the set formed from S by removing still spans H. b. If, some subset of S is a basis for H. REVIEW

Theorem The pivot columns of a matrix A form a basis for Col A. REVIEW

4.4 Coordinate Systems

Why is it useful to specify a basis for a vector space? One reason is that it imposes a “coordinate system” on the vector space. In this section we’ll see that if the basis contains n vectors, then the coordinate system will make the vector space act like R n.

Theorem: Unique Representation Theorem Suppose is a basis for V and is in V. Then For each in V, there exists a unique set of scalars such that.

Definition: Suppose is a basis for V and is in V. The coordinates of relative to the basis (the - coordinates of ) are the weights such that. If are the - coordinates of, then the vector in is the coordinate vector of relative to, or the - coordinate vector of.

Example: 1. Consider a basis for, where Find an x in such that. 2. For, find where is the standard basis for.

on standard basison

Example: For and, find. For, let. Then is equivalent to. :the change-of-coordinates matrix from to the standard basis

The Coordinate Mapping Theorem Let be a basis for a vector space V. Then the coordinate mapping is an one-to-one linear transformation from V onto.

Example: Let Determine if x is in H, and if it is, find the coordinate vector of x relative to.