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- Thread starter franky2727
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Mark44

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Do you mean A x A? If so, I don't think it could possibly be a subspace of the same vector space A is a subspace of.

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Mark44

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Here are the definition of a vector subspace and a very useful theorem, from wikipedia (http://en.wikipedia.org/wiki/Vector_subspace): [Broken]E Rand then the 3 questions have 3 different conditions (i) a+b=c (ii) ab=c+1 (iii) a^2=b^2 am i correct in thinking all this information is irrelivant as we have already been told that all 3 of these seperatly are E of R?

Let K be a field (such as the field of real numbers), and let V be a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. Suppose that W is a subset of V. If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V.

To use this definition, we don't have to prove that all the properties of a vector space hold for W. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following 3 conditions:

The zero vector, 0, is in W.

If u and v are elements of W, then the sum u + v is an element of W;

If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

Is the zero vector in each of the sets you listed? I can see that it is not in one of the ones you list, so that set is not a subspace (of R3). Are the other two conditions of the theorem satisfied for each set? I think I see one set for which one of the conditions doesn't hold.To use this definition, we don't have to prove that all the properties of a vector space hold for W. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following 3 conditions:

The zero vector, 0, is in W.

If u and v are elements of W, then the sum u + v is an element of W;

If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

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Dick

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