Section33supplementTimeless

# Section33supplementTimeless - Econ 204 Supplement to...

This preview shows pages 1–2. Sign up to view the full content.

Econ 204 Supplement to Section 3.3 The Matrix Representation of a Linear Transformation The purpose of this note is to provide a brief treatment of quotient vector spaces, and amplify on the relationship between linear transformations and their matrix representations. DeFnition 1 Given a vector space X and a vector subspace W X , deFne an equivalence relation by x y x y W Exercise 2 Show that is an equivalence relation. DeFnition 3 We deFne a new vector space V/W ,the quotient of V W . The set of vectors in is { [ x ]: x X } where we recall that [ x ] is the equivalence class of x with respect to the equivalence relation . In other words, [ x ]= { y X : x y W } = { x + w : w W } Thus, each of the vectors is a set ; this is a little weird at Frst, but try to get used to it. Now, we have to deFne the operations of vector addition and scalar multiplication. The deFnitions are [ x ]+[ y ]=[ x + y ] α [ x αx ] One needs to check that these deFnitions make sense. [ x ] is a set, and there are potentially many di±erent representatives, i.e. many x 0 such that [ x [ x 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at Berkeley.

### Page1 / 4

Section33supplementTimeless - Econ 204 Supplement to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online