(c) Let X C V be a lincar subspace such that X C Null(f), and let T: V > V/X be the natural surjcction T(v) = 7. (c. 1) Let f V/X W be given by f(7)f(v). (First shows that this is well defincd, that is, v, vi E V are such that w = vi (mod X), then (u) = T(v1), f() is independent of the choice of representative of .) Prove that f is a lincar transformation such that foT =f and f(V/X) = f(V) (c.2) Prove that if S: V/X -> W is a a lincar transformation such that SoT = f, then S f (c.3) Prove that f is 1-1 if and only if X isomorphic to Imag(f) {f(v): ve V}C W. SO Null(f). This proves that V/Null (f) is

(c) Let X C V be a lincar subspace such that X C Null(f), and let T: V > V/X be the natural surjcction T(v) = 7. (c. 1) Let f V/X W be given by f(7)f(v). (First shows that this is well defincd, that is, v, vi E V are such that w = vi (mod X), then (u) = T(v1), f() is independent of the choice of representative of .) Prove that f is a lincar transformation such that foT =f and f(V/X) = f(V) (c.2) Prove that if S: V/X -> W is a a lincar transformation such that SoT = f, then S f (c.3) Prove that f is 1-1 if and only if X isomorphic to Imag(f) {f(v): ve V}C W. SO Null(f). This proves that V/Null (f) is

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 4AEXP

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