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September 20th, 2012, 07:12 AM   #1
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Dimensions, linear transformations

Prove that if A: X-->Y is a linear transformation and V is a subspace of X then dimension of AV =< dimension of V. Deduce from here that rank (AB)=<rank B.
AV means the subspace V transformed by the transformation V, i.e any vector in AV can be represented as Av, v belonging to V.
I tried using the facts that AV is a subset of AX, V is a subset of X and i played around with the rank nullity theorem but could not prove it. Please help.
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September 21st, 2012, 08:22 AM   #2
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Re: Dimensions, linear transformations

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Originally Posted by amateurmathlover
Prove that if A: X-->Y is a linear transformation and V is a subspace of X then dimension of AV =< dimension of V. Deduce from here that rank (AB)=<rank B.
AV means the subspace V transformed by the transformation V, i.e any vector in AV can be represented as Av, v belonging to V.
I tried using the facts that AV is a subset of AX, V is a subset of X and i played around with the rank nullity theorem but could not prove it. Please help.
The rank nullity theorem that you reference says that dim V= dim(N)+ dim(I) where N is the nullity of A and I is its image. Look at the two cases, dim(N)= 0 and dim(N)> 0. All three numbers, of course, are non-negative.
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