# Linear algebra - subspaces

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I'm a bit confused about Linear spaces, particularly subspaces. One of the conditions of a vector space is that there is a zero vector such that x + 0 = x. For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off? Thanks :)

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 For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off?

You are correct. For a plane to be a subspace of R^3 it has to be of the form Ax+By+Cz=0 .

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The zero vector just says that p + 0 = p, which holds for both R^3 and R^2 so
(0,0,0) isn't necessary for the subspace to be R^2.

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Original post by Alrecenk
Quote:
 For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off?

You are correct. For a plane to be a subspace of R^3 it has to be of the form Ax+By+Cz=0 .

And likewise, a subset of the set of all polynomials would not be a subspace unless the zero polynomial was an element?

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 Original post by NerdInHisShoeAnd likewise, a subset of the set of all polynomials would not be a subspace unless the zero polynomial was an element?

Exactly. Keep in mind that the requirement of a zero element comes directly from the fact that a subspace is closed under scalar-vector multiplication, and the underlying field has a zero element. If x is in a subspace, then x*0 = 0 is in a subspace.

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If x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?

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 Original post by NerdInHisShoeIf x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?

That depends on whether the space's underlying field has infinitely many elements. In the real numbers, for example, the vector space would need to be infinite (except for the "trivial" vector space, as you said), but under some modular field (i.e. the integers modulo p), the space would not have to be infinite.

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 If x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?

Not necessarily. The underlying field may have a finite number of elements; for example, all Zp (that is, integers modulo p), where p is a prime, form a field. See Modular arithmetic, Galois field.

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It depends upon how you define + and * in your space. If you define them in the regular way then your space has to be infinite, but it's possible to define them such that that is not the case.

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