(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

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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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

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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