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Math / Linear Algebra 4 / Chapter 2.1 / Problem 13

Linear Algebra | 4th Edition | ISBN: 9780130084514 | Authors: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

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Let V and W be vector spaces, let T: V > W be linear, and let {w\,it)2, ,wk} be a

Chapter 2, Problem 13

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QUESTION:

Let V and W be vector spaces, let T: V > W be linear, and let {w\,it)2, ,wk} be a linearly independent subset of R(T). Prove that if S {vi,v2,... ,vk} is chosen so that T(vi) = Wi for i = 1,2,... ,k, then S is linearly independent

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QUESTION:

Let V and W be vector spaces, let T: V > W be linear, and let {w\,it)2, ,wk} be a linearly independent subset of R(T). Prove that if S {vi,v2,... ,vk} is chosen so that T(vi) = Wi for i = 1,2,... ,k, then S is linearly independent

ANSWER:

Step 1 of 2

Given transformation is linear, where and are two vector spaces.

Assume that the set is a linearly independent subset of and is a subset of and is a subset of such that for .

Consider for any set of scalars, consider the expression as;

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