The question states: "Prove that if $V \subset W$ and $dim(V) = dim(W)$, then $V = W$." (Both are subspaces with finite dimension) I am not sure how to prove this. I thought of proof by contradiction, i.e. supposing that there exists $x \in W$ such that $x$ isn't in $V$, but I couldn't really lead this into contradiction. My professor hinted me to start with a basis for $V$ and work from there, but I am not sure how this would lead anywhere.
Prove that if $V \subset W$ and $dim(V) = dim(W)$, then $V = W$
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linear-algebra
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0A basis of V extends to a basis of W. But because dim V= dim W , A basis of V is a basis of W. V=W. – 2017-02-22
1 Answers
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Take a basis for $V$, it contains $\dim(V)$ vectors that are linearly independant. Since $V \subset W$, these basis vectors are all in $W$, so we have $\dim(W)$ linearly independant vectors in $W$, so they form a basis. Now, any vector in $W$ can be written as a linear combination of these basis vectors, but this is necessarily in $V$, so we have that $W \subset V$ and hence $V=W$.