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If W is a subspace, is it also a linear space? If V is a linear space, is it also a subspace? I am having trouble wrapping my head around the difference between the two, as it seems that the way the book defines them is the following: both have to have a zero (neutral) element, both are closed under addition and scalar multiplication. Thanks!

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The main difference between refering to a vector spaces as a linear space or as a subspace is, unsurprisingly, context.

When one talks about a "subspace", one is thinking of it as being "inside" another vector space, with the inherited operations. So, if we refer to the set of all polynomials of degree at most $2$ with coefficients in $\mathbb{R}$ as a "subspace of the vector space of all polynomials" (note that subspace is relation term; something is a subspace of something else, thought the "something else" may be left implied or understood from context), then we are not merely thinking of this set as a vector space, we are thinking of it as a vector space sitting inside a larger vector space. On the other hand, if we refer to the set as "the vector space of all polynomials of degree at most $2$ with coefficients in $\mathbb{R}$", then we are not really interested in the fact that it is sitting inside a larger vector space, but we want to consider it as a vector space in and of itself.

We do that in set theory all the time, since a set may be a set or a subset of something else. The natural numbers form a set, and we refer to "the set of natural numbers" when we are focusing on the natural numbers exclusively; on the other hand, sometimes we want to think about the natural numbers and their relationship with the larger set of rationals or real numbers, so we can talk about "the subset of the real numbers consisting of the natural numbers."

Now, every vector space is a subspace of itself; every subspace of a vector space is itself a vector space (with the inherited operations). So whether we call them "space" or "subspace of <something>" is entirely a question of whether we want to focus on the object itself, or consider it in context with something else.

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I think some of your confusion my arise from being a bit imprecise in what you are saying. One generally says W is a subspace of V. To mean W is a subset of V containing 0 which is closed under the operations of addition and scalar multiplication which were defined for V.

Your second question (If V is a linear space is it also a subspace?) is a bit difficult to answer because you did not specify what V is a subspace of? Trivially, V is a subspace of itself, but I don't think this is what you meant.

Let's consider a geometric example. Consider the vector spaces $\mathbb{R}$ and $\mathbb{R}^{2}$ defined in the usual way. Intuitively, many people want to say $\mathbb{R}$ is a subspace of $\mathbb{R}^{2}$ but this is technically incorrect. we say instead that $\mathbb{R}$ is isomorphic to any 1-dimensional subspace of $\mathbb{R}^{2}$.

I hope this helps clarify the two concepts. As to why we bother with the concept of 2 different definitions, we can consider a set of homogeneous linear equations (things like 2x+3y+4z=0, the 0 on the RHS is what makes them homogeneous) in, say $\mathbb{R}^3$ and note that their solution set would form a subspace of $\mathbb{R}^3$

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Yes, a subspace is a space in itself. The conditions required for a subspace $+$ the conditions that the subspace inherits from the space make it a space in it's own right.

You should check here.

Why are they important? A subspace lives inside a space. The fact that it has in itself all the nice properties of a space make it interesting. For example, if you have a space $V$ and your subspace $W$ is generated by linearly independent vectors $v_1, v_2, ... , v_k$, then any vector that is a linear combination of these $k$ vectors also belongs to the same subspace.

(Sorry about those comments, pressed enter by accident)

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So there's no difference between the two? If not, what's the point of having two definitions? I hope you see where I am confused -- I see no difference. –  LinAlgStudent Feb 5 '11 at 21:00
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A subspace is a linear space which is part of a bigger linear space. –  Calle Feb 5 '11 at 21:09
    
Oh Calle, thanks a lot! I think that clarifies my confusion. :) So $R^2$ is a linear space, but it is also a subspace of itself, correct? –  LinAlgStudent Feb 5 '11 at 21:17
    
Yes yes yes yes. –  milcak Feb 5 '11 at 21:17

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