Expert: Paul Klarreich - 2/8/2006

Question
Hello I'm Ramy and I'm from Egypt... My Question is:

Let B be a BASIS of a vector space V and let W be a SUB-SPACE of V,then Let B' be the basis of W... If the DIMENSION of the Basis B is M and the DIMENSION of the Basis B' is N... then Can we always conclude that M=N .... I.E

Does the DIMENSION of the basis B'(of the sub-space) have to be always the same as the DIMENSION of the basis B(of the vector Space)?

Many thanks

Answer
Hi, Ramy,

Your Question:  Hello I'm Ramy and I'm from Egypt... My Question is:

Let B be a BASIS of a vector space V and let W be a SUB-SPACE of V,then Let B' be the basis of W... If the DIMENSION of the Basis B is M and the DIMENSION of the Basis B' is N... then Can we always conclude that M=N .... I.E

Does the DIMENSION of the basis B'(of the sub-space) have to be always the same as the DIMENSION of the basis B(of the vector Space)?????

Many thanks
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It has been a while since I did Linear Algebra, but I think the answer to your question here is

NO.  

If W is a subspace of V, then x in W implies x in V, but there COULD be things in V that are not in W.  

A 'basis', as I recall, is a set of vectors that generate (or span) the space.  So if B is a basis for V, then every x in V can be produced as a linear combination of elements of B.  You can have as many elements in your basis as you like, but you normally try to get a 'minimal' basis -- the smallest number of vectors necessary, and this number is the dimension of the SPACE.

So, for example, if V is the normal euclidean vector space of three dimensions, you must have at least three vectors for your basis.  It is common to take
(1,0,0), (0,1,0), and (0,0,1) as a basis, but you could find others.  All that is necessary is that you pick three vectors where no two of them could generate the third.  For example,
A(1,0,0), B(0,1,0), and C(2,3,0) do not make a basis for 3-space, because you can write C = 2A + 3B.

Now take W to be a plane in space, say the x-y plane, viewed as a subspace of the above 3-space.  This space can be generated by the A(1,0,0) and B(0,1,0) vectors listed before.  [NOTE: Since these are vectors in 3-space, we still write all three coordinates.]

So this subspace has a dimension of 2.