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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 )...
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