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This is a problem from our practice exam. Could anyone tell me how to approach this question and prove details.

Super appreciate!

Guess
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    The general idea for homework sort of stuff is that you make some effort, or at least outline where you are having difficulty. Also, the scan is a little hard to read mainly because of all the white space. – copper.hat Jul 25 '14 at 06:29
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1 Answers1

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a) An injective linear map always will preserve the dimension of the domain space, because it maps linearly independent set to linearly independent set. Therefore dim(im(T))=dim(domain space)=n where $m>n$.

b)Rank and Nullity theorem is a classical theorem in Linear algebra. dim(ker)+dim(im)=dim(domain space). T is onto, so dim(im)=m. From this you can deduce that dim(ker)=n-m where $n>m$

c) From $(a)$ and $(b)$ you can answer yourself the third question

Steven Van Geluwe
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Chellapillai
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