Hi, I have 4 implications I am interested in, I think I know the answer to the first 2, but the last two is not something I know, however they are related to the first 2 so I will include all to be sure.(adsbygoogle = window.adsbygoogle || []).push({});

Assume that T is a linear transformation from from vectorspace A to B.

T: A -> B

A* is n vectors in A, that is A* = {a1, a2, an}

1.

T(A*) linearly independent -> A* linearly independent

If T(A*) is linearly independent, then A* must be linearly independent, without any requirements for T?

2.

T(A*) lindearly dependent -> A* linearly dependent

If T(A*) is linearly dependent, then we can only conclude that A* is linearly independent only if T is 1-1, if T is not 1-1 we can not conlude anything?

3.

span(T(A*))=B -> span(A*)= A

If span(T(A*)) = B, what requirement must we have to conlude that span(A*) = A. That T is 1-1, surejective, both or none?

4.

span(T(A*)) ≠ B -> span(A*) ≠ A*

If span(T(A*)) is not B what must we have to conlude that span(A*) is not A? Will this implication hold if T is 1-1, surjective, both or none?

thanks

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# Linear algebra, when does the implications hold?

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