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If A and B are vector spaces such that

A = B

Do they have the same dimension?

I think yes, because if they are equal then they can be spanned by some vectors. Any vectors that span A span B. Suppose then that A has a smaller dimension, meaning that it can be spanned by a smaller number of linearly independent vectors. Well, if these vectors span A, then they must be able to span B as well meaning that the dimensions are equal. Thus, A cannot have a smaller dimension (different dimension) than B.

Let me know if my logic is incorrect.

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    YES! Thank you.2012-11-12

1 Answers 1

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Yes, correct, but the question seems too trivial.

Moreover, try to prove that there is a bijective linear mapping between vector spaces $A$ and $B$ if and only if $\dim A=\dim B$.

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    The original exercise was not in the question, I just want to make sure that a n x n matrix A where nullspace(A) is equal to colspace(A) is not invertible. This is because if the null space and the column space were the same, they'd have the same dimension. Which means that the nullity(A) = dim[colspace(A)] = rank(A). By the Rank-Nullity Thm., rank(A) + nullity(A) = the number of columns (and in this case, rows). Which means that n = 2*rank(A) which means that there are some free variables which means that the matrix is not invertible. Does that make sense?2012-11-12