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I'm reading this Wikipedia article on sequences Sequence where it mentions 'Sequences over a field may also be viewed as vectors in a vector space.' under vectors section. I'm not able to grasp this, can somebody explain how vectors can be defined in terms of sequences?

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2 Answers 2

The point is: "How do think of 'vectors'?" or "What is a 'vector'?" Here a vector is just seen as an element of a vector space, not as a $n$-tuple of elements from the ground field. The set \[\sideset{^{\mathbb N}}{} F = \{(a_n) \mid a_n \in F, n \in \mathbb N\} \]of sequences of elements from a field $F$ is a vector space over $F$, if we define the vector space operations

  • Addition The sum of two sequnecens $(a_n), (b_n) \in \sideset{^{\mathbb N}}{} F$ is defined as the sequence $(a_n + b_n)$
  • Scalar multiplication The product of $\lambda \in F$ with a sequence $(a_n) \in \sideset{^{\mathbb N}}{} F$ is defined as $(\lambda \cdot a_n)$.

Then all properties of a vector space are fulfilled ($\sideset{^{\mathbb N}}{} F$ with $+$ is an abelian group, and we have associativity of the multiplications, distributivity...). That is, sequences are more or less a vector space in the same way finite tuples are, by elementwise operations.

As the sequences form a vector space, we may call them vectors.

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In mathematics vector spaces are defined formally by being able to add in them and scalar multiply with an element of the field (for details see the corresponding Wikipedia article).

You just need to figure out, what both of them is for sequences, but that is not really a problem.

  • Addition: Let $(a_n)$ and $(b_n)$ be sequences. We define the addition as follows: $$(a_n) + (b_n) := (c_n) \text{ with } \forall n\in\Bbb N\colon c_n := a_n + b_n$$
  • Multiplication: Let $(a_n)$ be a sequence and $\lambda$ an element of the underlying field (i.e. $\Bbb R$). Then the multiplication is defined as: $$ \lambda (a_n) := (c_n) \text{ with } \forall n\in\Bbb N\colon c_n := \lambda a_n$$

You just need to check all the properties from the definition (like associativity etc.) to see that this indeed defines a vector space, however of infinite dimension.

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