There is a problem with an answer, however I don't understand why there is any need for the bolded part? Isn't it superfluous?

Thanks in advance!

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- Jan 2nd 2013, 09:41 AMjojo7777777nonsingular linear map- help with some proof
There is a problem with an answer, however I don't understand why there is any need for the bolded part? Isn't it superfluous?

Thanks in advance! - Jan 2nd 2013, 11:31 AMDevenoRe: nonsingular linear map- help with some proof
it's an "if and only if" proof.

the "if part": if h preserves linear independence, h is non-singular. this is the part that is proved by the bold section. this is actually proved by proving the contra-positive: h is singular implies h takes SOME linearly independent set to a linearly dependent one.

the linearly independent set they choose is S = {v}, where v is some non-zero element of the null space (which exists BECAUSE h is singular). this of course, maps to h(S) = {0_{W}}, which is ALWAYS linearly dependent.

you have to include BOTH implications in an if and only if proof, you can't leave one out. - Jan 2nd 2013, 12:16 PMjojo7777777Re: nonsingular linear map- help with some proof
Thank you very much...!!!