Question

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Let A ∈ Mm,n(F) and let TA : F n → F m be the corresponding linear map (see (1.2)). (a) Prove that A has a left inverse if and only if TA is injective. (b) Prove that A has a right inverse if and only if TA is surjective. (c) Prove that A has a two-sided inverse if and only if TA is an isomorphism.

Answer 1

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Answer:

Explanation:

(a) Suppose that has a left inverse ; then is the identity matrix. If such that then

Thus, is injective.

Conversely, suppose that is injective. Then the map is a linear isomorphism. Let be the standard basis of ; then the image is a basis of the image ​​; extend this to a basis of . Define as follows:

Let be the matrix representing , so that . Then is matrix such that

for all . Thus, , which means has a left inverse .

b) Suppose that has a right inverse ; then is the identity matrix. If then and we get

Thus, is surjective.

Conversely, suppose that is surjective. Let be the standard basis of . Let be such that for all ; then the set is linearly independent in , and therefore, can be extended to a basis of . Define as follows: .

Let be the matrix representing , so that . Then is matrix such that

for all . Thus, , which means has a right inverse .

c) Suppose that has a left inverse and a right inverse . Then, by part a) and b), the map is injective and surjective. Hence, it is bijective. Since a bijective linear transformation is an isomorphism, we conclude that is linear isomorphism.

Conversely, suppose that is linear isomorphism. Then it is bijective, hence, both injective and surjective. By parts a) and b) the matrix has a left inverse and a right inverse . But then . Thus, is a two-sided inverse of .

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