Lect22

# Lect22 - Chapter 4 Linear Transformations Math1111 Matrix...

This preview shows pages 1–7. Sign up to view the full content.

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

View Full Document

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

View Full Document

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.

Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Construction of Linear Transformation Theorem 4.2.1 says that a linear transformation L : R n → R m is represented by an m × n matrix A , which is constructed from the standard basis e 1 , ··· , e n for R n . Theorem Let V and W be vector spaces. Suppose v 1 , ··· , v n forms a basis for V . Let z 1 , ··· , z n be (not necessarily distinct) vectors in W . Then there is a unique linear transformation T : V → W such that T ( v j ) = z j . The linear transformation T is obtained by linear extension . Example . Find a linear transformation T : R 2 → R 3 such that T ( e 1 ) = ( 1 1 1 ) T , T ( e 2 ) = ( 2 1 ) T . Chapter 4. Linear Transformations Math1111 Matrix Representations Construction of Linear Transformation Theorem 4.2.1 says that a linear transformation L : R n → R m is represented by an m × n matrix A , which is constructed from the standard basis e 1 , ··· , e n for R n . Theorem Let V and W be vector spaces. Suppose v 1 , ··· , v n forms a basis for V . Let z 1 , ··· , z n be (not necessarily distinct) vectors in W . Then there is a unique linear transformation T : V → W such that T ( v j ) = z j . The linear transformation T is obtained by linear extension . Example . Find a linear transformation T : R 2 → R 3 such that T ( e 1 ) = ( 1 1 1 ) T , T ( e 2 ) = ( 2 1 ) T . Ans . T (( x y ) T ) = ( x + 2 y x x + y ) T . Chapter 4. Linear Transformations Math1111 Matrix Representations Construction - Proof Proof of Theorem For every v ∈ V , v = α 1 v 1 + α 2 v 2 + ··· + α n v n . Define T : V → W by T ( v ) = α 1 z 1 + α 2 z 2 + ··· + α n z n . Check that (i) T is a linear transformation. (ii) If S : V → W is a linear transformation such that S ( v j ) = z j for j = 1,2, ··· , n , then S = T . Chapter 4. Linear Transformations Math1111 Matrix Representations Construction - Example Example . Is there a linear transformation T : R 3 → R 2 such that T ( u 1 ) = ( 1 1 ) T , T ( u 2 ) = ( 1 1 ) T , T ( u 3 ) = ( ) T where u 1 = ( 1 1 ) T , u 2 = ( 1 ) T , u 3 = ( 1 ) T ? If yes, describe it and find the standard matrix representation. Chapter 4. Linear Transformations Math1111 Matrix Representations Construction - Example Example . Is there a linear transformation T : R 3 → R 2 such that T ( u 1 ) = ( 1 1 ) T , T ( u 2 ) = ( 1 1 ) T , T ( u 3 ) = ( ) T where u 1 = ( 1 1 ) T , u 2 = ( 1 ) T , u 3 = ( 1 ) T ? If yes, describe it and find the standard matrix representation. Ans . Observe that u 1 , u 2 , u 3 forms a basis for R 3 . We extend linearly to define the linear transformation, i.e. define T x 1 u 1 + x 2 u 2 + x 3 u 3 = x 1 T ( u 1 ) + x 2 T ( u 2 ) + x 3 T ( u 3 ) . Chapter 4. Linear Transformations Math1111 Matrix Representations Construction - Example Example . Is there a linear transformation T : R 3 → R 2 such that T ( u 1 ) = ( 1 1 ) T , T ( u 2 ) = ( 1 1 ) T , T ( u 3 ) = ( ) T where u 1 = ( 1 1 ) T , u 2 = ( 1 )...
View Full Document

## This document was uploaded on 05/04/2011.

### Page1 / 32

Lect22 - Chapter 4 Linear Transformations Math1111 Matrix...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online