What is the difference between Set of Vectors and Sequence of vectors?

Question

This question was recently asked tn My University's exam on Linear Algebra University Exam When I saw the question I thought I knew it But later I realised that I don't completely understand the difference between them.

Answer 1

A set of vectors is just a subset of a vector space $V$ while a sequence of vectors is a map $\mathbb{N} \rightarrow V$ (can also be written as a infinite tuple). A set does not care about ordering or enumerating elements multiple times in contrast to a sequence.

Answer 2

In a set items cannot appear more than once whereas in a sequence they can. Items in sequences are ranked in order whereas there is no order for sets.

Answer 3

A set is a collection of things in no particular order, with no duplicates allowed. Often the elements are written between braces, so $$ \{7,2, 3, 5, 3\} = \{2,3,5,7\}. $$

A sequence is a collection of things in order, with repetitions allowed. It's often written between parentheses, so $$ (7,2, 3, 5, 3) \ne (2, 3, 3, 5, 7). $$

You have to be a little more careful with notation for infinite sets and infinite sequences but the idea is the same.

Sometimes people are a little careless when the distinction isn't crucial. If you edit the question to tell us the precise statement of the exam question we can parse it for you.

Answer 4

In general, a set and a sequence differs in two ways:

1. The order among the elements in a sequence is important, while that of the elements in a set is not.

2. An element may be repeated in a sequence, but it does not make sense to repeat an element in a set, because a particular element is either in the set, or not in the set.

For example, consider the sequences $1,2,3,1,2,3,1,2,3,\dots$, and $1,3,2,1,3,2,1,3,2,\dots$. While the two sequences comprise of the same set of elements $\{1,2,3\}$, they are distinct sequences, because the same elements occur at distinct indices in the two sequences. On the other hand , $\{1,2,3\}$, $\{2,1,3\}$, and $\{1,2,3,3,2,2\}$ are all the same set, because neither the order nor the multiplicities of elements matter for a set.