4 Vector Spaces 4 1 Vector Spaces and

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 REVIEW 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.

The Spanning Set Theorem Let REVIEW 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. .

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

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

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

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

on standard basis on

Example: For Then and , find , let . . 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.