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a) Prove that if linear transformation T: U → V and S: V → W, then the composition (S.T) : U -> W is also a linear transformation.

a) Prove that if linear transformation T: U → V and S: V → W, then the composition (S.T) : U -> W is also a linear transformation.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 43EQ
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a) Prove that if linear transformation T: U → V and S : V → W, then the composition (S.T): U -> W is also a linear transformation. b) Find the matrix transformation R2 • R,• 2 Id : R2 → R2 which is a `y composition of 2•ld – scales a vector in R2 by a factor of 2, R, reflects a vector about the y-axis and R12 – rotates a vector by angle T/2 in a counter clockwise direction. c) Prove that if T is an injective linear transformation i.e T : U –→ V is linear, then dim U (dimensions U) is less than or equal to dim V (dimensions V).
Transcribed Image Text:a) Prove that if linear transformation T: U → V and S : V → W, then the composition (S.T): U -> W is also a linear transformation. b) Find the matrix transformation R2 • R,• 2 Id : R2 → R2 which is a `y composition of 2•ld – scales a vector in R2 by a factor of 2, R, reflects a vector about the y-axis and R12 – rotates a vector by angle T/2 in a counter clockwise direction. c) Prove that if T is an injective linear transformation i.e T : U –→ V is linear, then dim U (dimensions U) is less than or equal to dim V (dimensions V).
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