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When a vector space is just a set of vectors just like any other linear space . Then why is it that you always need a basis to express vectors ?

For example , you don't generally need a basis to express the elements of the set of real numbers , like you don't see someone expressing (4) as '4' x (1) always. Where (1) is the basis and '4' is the scalar and (4) is the element of linear space of real numbers .

but for vectors you need to resolve the vector into components of the basis . Why ?

And why is this thing called a " Space " ?

And I think I understand what a number is , but what is a vector ? Or in fact what is even a number ?

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2 Answers 2

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You think you always need a basis because you do not know enough examples. Perhaps the definition of a vector space over $\mathbb R$ would make things clearer.

A set $V$ together with operations $+ : V \times V \to V$ and $\cdot : \mathbb R \times V \to V$ is called a vector space over $\mathbb R$ if they satisfy these properties : $$ \forall a,b,c \in V, \quad a+b = b+a, \quad a+(b+c) = (a+b)+c, $$ and there exists $0 \in V$ such that $\forall a \in V, a+0 = a$. Furthermore, for each $v \in V$, there exists another vector called $-v$ such that $v+(-v) = 0$. For the scalar multiplication, we require $$ 1 \cdot v = v, \qquad r \cdot (s \cdot v) = (rs) \cdot v, $$ and distributivity from both sides : $$ r \cdot (v_1 + v_2) = r \cdot v_1 + r \cdot v_2, \qquad (r_1 + r_2) \cdot v = r_1 \cdot v + r_2 \cdot v. $$ If you have something like this, it's called a vector space. You don't need a basis to express its elements. For instance, if you take the space of all functions from $\mathbb R \to \mathbb R$, they can be added ($(f+g)(x) = f(x) + g(x)$ is the definition of the sum of two functions), they all have inverses (the zero of this space would be the $0$ function) and multiplication by scalar is defined as $(\lambda f)(x) = \lambda (f(x))$. The properties are all satisfied, hence you have a vector space.

Note that this example is infinite dimensional (in the simple sense that it doesn't admit a finite basis). In infinite dimension, the discussion of bases is less trivial than in finite dimension.

A number usually stands for an element of $\mathbb R$, but for some people a number can also mean a complex number (i.e. an element of $\mathbb C$), so you need to notice the context. A vector stands for an element of a vector space, so saying that something is a vector refers to the vector space it lives in.

Hope that helps,

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Intuitively , a vector space is a set of vectors and a vector is an element of vector space ? Isn't this circular ? Or am I missing something ? And same for the numbers just like vectors . –  nonagon May 6 '13 at 6:15
    
Intuitively, you need examples to build your intuition, but a vector space is just a set with two operations satisfying some properties. It need not look like $\mathbb R^n$, it can be wayyyy "bigger". Saying that "a vector space is a set of vectors" is not right, in the sense that the definition of a vector space is the one written above, and the elements of the set $V$ are called vectors of the vector space in this case. A vector space is not simply a set of vectors, there is some structure on it. –  Patrick Da Silva May 6 '13 at 6:30
    
How do you arrive at the conclusion that nonagon doesn't know enough examples? –  dezign May 6 '13 at 23:49
    
@dezign : Thinking that a vector space always needs a basis to write its elements is not true, so I believed he did not know examples of vector spaces over which decomposing over the Hamel basis is not a nice way to write the elements of the space. –  Patrick Da Silva May 6 '13 at 23:54

You don't have to use bases, they just make your life easier when dealing with vector spaces. Since $\mathbb{R}$ is one dimensional, the notion of basis is not that useful. As you pointed out, you don't gain much from writing $4$ as $4\cdot 1$, taking $1$ as a basis of $\mathbb{R}$. More generally, when dealing with some sort of structure or collection of objects it is useful to decompose the structure or collection of objects into smaller more fundamental components (or "building blocks"), such that all the elements of your structure or collection are just certain combinations of these more fundamental building blocks (for a non-mathematical analogy think of say the periodic table of elements in chemistry). So even though finite dimensional vector spaces over the real or complex numbers have uncountably many elements, the situation is simplified dramatically once we can identify a linearly independent set of vectors which span the space, then we know that all other vectors are just certain combinations of these fundamental "building blocks".

As for why vector spaces are called spaces, it's just because once a basis is chosen we can think of them in a geometric way just as we do with $\mathbb{R}^n$ or $\mathbb{C}^n$. You don't have to think of vector spacers as spaces if you don't want, but if you like to use your visual sense at all when thinking of mathematical structures (as I do) then thinking of vector spaces as actual "spaces" enables one to use their geometric intuition along with the algebraic formalism.

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Bases don't always make your life easier. I don't think you wanna use a Hamel basis over $\mathbb R^{\mathbb R}$ for instance. –  Patrick Da Silva May 6 '13 at 23:55
    
Maybe the OP will feel better about himself after reading your answer, but you're just confusing him. The only reason we call a space a space is purely arbitrary, it's a pretty word, that's it. We would've called a group an "algebraic space" if we would've wanted to, but the first mathematicians that started using groups ended up calling them this way, that's it. The name vector space must've come up after many mathematicians called those things "spaces of vectors", i.e. somewhere where vectors lived. There's really nothing mathematical about the word "space". –  Patrick Da Silva May 6 '13 at 23:58
    
And bases are really not that useful once you leave the context of Hilbert spaces, so my answer pointed in that direction, since OP said he "always needed a basis to write the vectors", and I said no, this is not true. You're tricking him into thinking he will always somehow be writing an element as if he knew the underlying basis... –  Patrick Da Silva May 7 '13 at 0:00
    
So you have not only come to the conclusion that the OP doesn't know enough examples, but that I am confusing him even though he accepted my answer. I'm sure on the contrary that answering his question by giving the definition of a vector space really cleared things up for him though. –  dezign May 7 '13 at 0:03
    
Yes, and I stand by what I said. There is a difference between being accepted on MSE and trying to give the OP the right picture. Don't take it personal ; I just think that OP's question is quite abstract and giving him handwaving is just telling him to give up on understanding really what's going on. If he accepted your answer I'll just guess he followed your tip instead of mine and that's fine with me. –  Patrick Da Silva May 7 '13 at 0:58

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