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Math / Linear Algebra: A Geometric Approach 2 / Chapter 4.4 / Problem 18

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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a. Suppose T : V W is a linear transformation. Suppose {v1, . . . , vk} V is linearly

Chapter 4, Problem 18

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QUESTION:

a. Suppose T : V W is a linear transformation. Suppose {v1, . . . , vk} V is linearly dependent. Prove that {T (v1), . . . , T (vk)} W is linearly dependent. b. Suppose T : V V is a linear transformation and V is finite-dimensional. Suppose image (T ) = V . Prove that if {v1, . . . , vk} V is linearly independent, then {T (v1), . . . , T (vk)} is linearly independent. (Hint: Use Exercise 12 or Exercise 16.)

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QUESTION:

a. Suppose T : V W is a linear transformation. Suppose {v1, . . . , vk} V is linearly dependent. Prove that {T (v1), . . . , T (vk)} W is linearly dependent. b. Suppose T : V V is a linear transformation and V is finite-dimensional. Suppose image (T ) = V . Prove that if {v1, . . . , vk} V is linearly independent, then {T (v1), . . . , T (vk)} is linearly independent. (Hint: Use Exercise 12 or Exercise 16.)

ANSWER:

Step 1 of 4

a. Show that if the set is linearly dependent, then the set is also linearly dependent

It is known that is a linear transformation and the set in is linearly dependent.

By definition of linear dependence, there exists not all zero constants such that

.

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