MATH 323 Lecture 17

From Notes
Jump to navigation Jump to search

« previous | Thursday, October 25, 2012 | next »

Linear Transformation

Rotation Transformation

Let be a rotation by angle about the origin.

In General

For arbitrary , where is a basis in and is a basis in .

is the coordinate vector of w.r.t so

For some , we can represent as .

[1], so what is ?


Theorem 4.2.2

Matrix representation theorem

If and are ordered bases for vector spaces and respectively, then corresponding to each linear transformation there is an matrix such that

for each

is the matrix representing relative to the ordered bases and .

In fact, , where for .


For and is a basis in , find the matrix representing w.r.t. ordered bases (standard 3D basis) to

Just take

, where is a basis for . Find .

, where .

forms basis for , and forms basis for


Reversing Linear Transformations

Theorem 4.2.3

Given and are bases for and respectively,

If is the matrix representing w.r.t. and , then

for , where

Corollary 4.2.4

If is the matrix representing the linear transformation w.r.t. and , then the rref of is .



Basis is ,

Basis is , ,

What is w.r.t. and ?


  1. The correspondence between and given by ; and between and is called isomorphism