Please, help me with very detailed and step by step solutions for my understanding of questions c1, c2, and c3. I will be much appreciative of your solutions. Thank you This question is about Quotient space: Let V and W be vector spaces over a field K and f:V→W a linear map. (c) Let X C V be a lincar subspace such that X C Null(f), and let T: V > V/X bethe natural surjcction T(v) = 7.(c. 1) Let f V/X W be given by f(7)f(v). (First shows that this is welldefincd, 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 lincartransformation 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 Xisomorphic to Imag(f) {f(v): ve V}C W.SONull(f). This proves that V/Null (f) is

(i) The idea is to factor the given linear map f from V to W, using the natural quotient map t( tau)…

Answered: (c) Let X C V be a lincar subspace such…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please, help me with very detailed and step by step solutions for my understanding of questions c1, c2, and c3. I will be much appreciative of your solutions. Thank you

This question is about Quotient space: Let V and W be vector spaces over a field K and f:V→W a linear map.

(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

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