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Do all vector spaces have standard bases?

I was reading a book which says that "it makes no sense talking about standard bases for any vector space". I don't really understand this concept.

If this is right, under what conditions does a vector space have a standard basis?

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    Note that given a vector space, there is not criterion saying which basis is best. Given a problem, you just pick the basis which you like, which makes computations easier (for example a orthonormal basis). Given any base, using the change of basis matrix, one can switch between bases. – 2017-01-30
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    i know this but what about standard basis? maybe my doubts are on the differences bewteen standard basis and orthonormal basis – 2017-01-30
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    The term 'standard basis' is just used to point at that there is a certain choice of 'default basis': in $\mathbb{R}^n$ we let the default basis be the one with vectors $(1, 0, \ldots, 0), (0, 1, \ldots, 0), \ldots, (0, 0, \ldots, 1)$. This happens to be a orthornormal basis (each vector is orthogonal to the others and has lenght 1). However, this is just a choice and since you can always switch between bases, the question 'under what conditions a vector space has a standard basis' is not usefull. – 2017-01-30
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    you're right, but you're talking about Rn. may exist any vector spaces which don't have standard basis? – 2017-01-30
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    It is just a name... People call the basis i wrote down for $R^n$ the standard basis to avoid writing it down every time. It is just a name and nothing more. – 2017-01-30
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    so it's just a problem of name? from it, seems that is not always possibile find a basis with vectors (1,0,...,0), (0,1,...,0), ..., (0,0,...,1) – 2017-01-30
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    Given any basis in $\mathbb{R}^n$, you can find a matrix transforming this basis to the one you wrote. – 2017-01-30
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    It is merely a name, so if a vectorspace has a basis which people call the standard one, then given any other basis you can move to the 'standard one' and vice versa. – 2017-01-30
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    This is my problem. it's possible that a vector space doesn't have the "standard one"? – 2017-01-30
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    Look: the term 'standard basis' is just a name. People agreed that they would call the basis $(1, \ldots, 0), \ldots, (0, \ldots, 1)$ the standard basis for $\mathbb{R}^n$ in order to make it easy to talk about. It does not make sense to ask whether or not 'there is a standardbasis' since it does not matter: if you feel like you like the basis $(1, \ldots, 0), (1,1, \ldots, 0), \ldots, (1,1,\ldots, 1)$, you could call it 'my favourite basis'. However, people liking 'the standardbasis' would still be able to make computations by just moving from your 'favourite basis' to their standard basis – 2017-01-30
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    why do you even care that much about this standard basis? you could literally use every basis! Let me ask a different question: what would be the your specific problem if there is some vectorspace without a standard basis? – 2017-01-30
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    i'm studying for an exam and i messed up with this affirmation made by my professor. i'm sorry, but surely I'm not doing the right question because of the name issue. i've doubts with the basis (1,…,0),…,(0,…,1) and not with its name – 2017-01-30
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    What is your doubt about that basis? – 2017-01-30
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    i just want to know if every vector spaces have the basis (1,…,0),…,(0,…,1). – 2017-01-30
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    Did my answer help you? if not feel free to ask any more questions. If so, would you consider accepting it by clicking on the check mark? (Since I'm only a beginner, i could use the experency). – 2017-01-30
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    do you believe that every vector spaces have the basis (1,...,0),...,(0,...,1)? – 2017-01-30
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    look at my answer pleas. It does not make sense to write that a basis of $P_n$ woudl be the one you write down. However, if we consider the coordinates, we see that the set you wrote down consists out of the coordinates of the chosen basis vectors with respect to that chosen basis. – 2017-01-30

2 Answers 2

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Consider the vector space consisting out of real polynomials of degree less than or equal to $n-1$. That is: $$P_{n-1} = \{a_{n-1}x^{n-1} + \ldots + a_1x + a_0\}.$$ A basis for this vectorspace is given by $$\{1, x, x^2, \ldots, x^{n-1}\}$$ (check this). Now any polynomial of degree less than or equal to $n-1$ is uniquely determined by the coefficients with respect to this basis: for example the polynomial $x^2 + x -1$ in $P_3$ has coefficients $(-1, 1, 1, 0)$. (Do you see this?) But we can also describe the basis elements in coordinates with respect to this basis: for example $$x = 0 \cdot 1 + 1 \cdot x + 0 \cdot x^2 + 0 \cdot x^3$$ in the case of $P_3$. written as a vector: $(0,1,0,0)$.

This is what we need to when we want to describe the matrix corresponding to some linear transformation: suppose $V$ and $W$ are vectorspaces of dimension $n$ respectively $m$. We want to describe the matrix corresponding to some linear transformation $L: V \to W: v \mapsto L(v)$. In order to be able to do this, we need to have bases of $V$ and $W$. This allows us to describe all elements of $V$ respectively $W$ using their coordinates with respect to the chosen basis of $V$ respectively the chosen basis of $W$.After we have choosen this basis, we find that the coordinates of the basisvectors with respect to the basisvector are exactly the vectors $(1, 0, \ldots, 0)$ etc. I hope this solves your problem lot.

NOTE: although this does not seem to answer the original question of the OP, I have posted this answer in the light of our discussion in the comments on this question. This answer however was way too long to post as a comment.

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A vector space has a standard basis if and only if people have selected one particular basis and given it the name "standard". In the case of $\mathbb R^n$, people have done that, with the basis consisting of vectors with a single component equal to $1$ and all other components $0$. Furthermore, this particular choice has been so widely accepted that you can say "standard basis for $\mathbb R^n$ and reasonably expect all mathematicians to understand what you mean.

For other vector spaces, someone might choose a basis and call it standard, especially if this happens in the context of a paper in which that basis is particularly useful. But there might be no general agreement among mathematicians that this particular basis deserves the name "standard". So it might be called "the standard basis" in a particular paper, but not in other contexts. (Furthermore, different people, working in different contexts, might have proposed different bases as "standard" for the same space.)

There are also plenty of vector spaces for which no one has (yet) selected a particular basis and named it "standard".

I would use the phrase "standard basis" only in two situations: (1) Cases like $\mathbb R^n$ where we all agree as to what the standard basis is, so I could refer to it without fear of confusion. (2) Cases where I've chosen a particular basis, decided to call it "standard", and explicitly said (earlier in the same paper) which basis I meant by "standard basis".