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.
How to show that two vector spaces $V$ and $W$ are the same
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vector-spaces
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3You should call the vector spaces something other than $n$ and $m$, if you wish to use those symbols for the dimensions. – 2012-02-07
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2Not 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. – 2012-02-07
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0Thanks guys would keep that in mind for future questions. – 2012-02-07
1 Answers
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Assuming the dimensions are finite, show that a basis of $V$ is a basis of $W$.
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0As 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 ? – 2012-02-07
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1@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 $V=\text{span}{\frak B}=W$. – 2012-02-07
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0Cool, 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 ? – 2012-02-07