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How can sequences satisfy vector space properties? If sequence is not a vector.

An example semantic problem occurs, when one needs to find a zero element or zero $\bar{0}$. In order to do this for sequences one'd intuitively define $\{a_n\}, a_n=0 \forall n$ as the zero element. However, this is a sequence, not a vector.

As to why the treating of sequences as vectors may not be intuitive:

What is the difference between operators, functions, sequences and vectors?

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    $\begingroup$ This is too vague. One sequence is just a single object, so it can't be a vector space (or at least not a very interesting one). It could, of course, be an element in a vector space. $\endgroup$ – lulu Jan 11 at 11:00
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    $\begingroup$ Sure they do. $\{a_n\}+\{b_n\}=\{a_n+b_n\}$ and $\lambda \times \{a_n\}=\{\lambda a_n\}$. $\endgroup$ – lulu Jan 11 at 11:02
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    $\begingroup$ i just wrote out the formal definition. Where's the problem? $\endgroup$ – lulu Jan 11 at 11:03
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    $\begingroup$ I'm really not sure what you are after here. If you have a field $\mathbb F$, then the set of sequences of elements in $\mathbb F$ form a vector space in an obvious way. Any such sequence is then, of course, a vector in that vector space. $\endgroup$ – lulu Jan 11 at 11:07
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    $\begingroup$ An element of a vector space is called a vector. It may or may not have any resemblance to 'vectors' $\endgroup$ – Shubham Johri Jan 11 at 11:13
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You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.

Consider the following passage from that answer:

If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.

The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see) give only limited perspectives into the field of mathematics. It is likely that someone will see only finite-dimensional vector spaces, or at least that only the finite-dimensional vector spaces will be identified as vector spaces.

The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors. There are various ways to state a formal definition of vector space (here is an example), but they generally admit the same models of vector spaces, for example sequences of elements of a field $F$ as described in the passage above.

This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space until applications like this become intuitive.

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  • $\begingroup$ The confusion may not come from "elementary mathematics" but instead from "physics", where the term vector is used with their special connotations. Even in more advanced physics, when they use a Hilbert space, its elements are not called "vectors" but something else, such as "states". $\endgroup$ – GEdgar Jan 11 at 13:09
  • $\begingroup$ @GEdgar True, and reading even further in the discussion of the earlier question, even some fields of higher mathematics differ in how they use certain terms. So it really would depend on which communities one has been exposed to. On the other hand, after one has seen definitions get broadened or narrowed a few times I suspect it gets easier to get past stumbling blocks such as the one we seem to have here (whatever its exact nature is). $\endgroup$ – David K Jan 11 at 13:27
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A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.

In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.

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