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How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help would be much appreciated.

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You should call the vector spaces something other than $n$ and $m$, if you wish to use those symbols for the dimensions. – David Mitra Feb 7 '12 at 18:57
Not true if they are infinite dimensional. And, as David said, don't use $m$ and $n$ for vector spaces, use capital letters, often $V$ and $W$ are used. – Thomas Andrews Feb 7 '12 at 19:02
Thanks guys would keep that in mind for future questions. – Comic Book Guy Feb 7 '12 at 19:07
up vote 9 down vote accepted

Assuming the dimensions are finite, show that a basis of $V$ is a basis of $W$.

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As per definition of a basis in this case every vector in V can be expressed using V's basis vectors and every vector in W can be expressed using W's basis vectors and since V is a subspace of W all vectors in V can be expressed using W's basis vectors. We also know the dim V = dim W so can we conclude from that W's basis is also a basis of V and hence V = W ? – Comic Book Guy Feb 7 '12 at 19:05
@Hardy Do not worry about a basis of $W$. Start with a basis ${\frak B}$ of $V$. Then show it is a basis of $W$. Once you've done this, you'll know $V=\text{span}{\frak B}=W$. – David Mitra Feb 7 '12 at 19:29
Cool, thanks for that tip so a basis B = {x1, x2 ... xn} of V can span all of V, but since W is also of the same dimension hence the same basis B can span all of W as well as the subspace must be because of the restriction on the scalar coefficients of V's vectors ? Now sure if that was the best way to express it though ? – Comic Book Guy Feb 7 '12 at 19:35

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