This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full 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 . 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 . Allusion A linear transformation L is determined by the images/outputs of the basis in the domain. 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 . 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  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 ?...
View
Full Document
 Spring '10
 Dr,Li
 Math, Linear Algebra, Transformations, Vector Space, linear transformation, Matrix Representations

Click to edit the document details