Question

Complete the following questions. (a) Which two conditions need to be satisfied for a set of vectors in a vector space to form a subspace. (b) According to part (a), determine whether all sequences ū in Roo of the form ū = (v,1,0,1,0,1,...) form a subspace.

Answer 1

(a) Let
`U\subset V` be a subset of a vector space
`V`.
`U` is a subspace of
`V` if and only if the following two conditions hold:

i)
`U` is closed under the sum operation. That is to say,
`u_(1)+u_(2)\in U` whenever
`u_(1),u_(2)` are elements of
`U`.

ii)
`U` is closed under the scalar multiplication. That is to say,
`\lambda u\in U` whenever
`u\in U` and
`\lambda \in \mathbb{R}`

Explanation:

For the part (b) we have the set of all sequences of the form
`\bar{u}=(v,1,0,1,0,1,...)`, where
`v\in \mathbb{R}`. Observe the if you multiply any sequence of this form by and scalar
`\lambda\\eq 1` then the sequence stops being like the given form. For example, let
`\lambda=5`. Then:

`5 \bar{u}=(5v,5,0,5,0,5,...)`

This implies that the set under consideration is not closet under scalar multiplication, which implies that the set is not a subspace of the vector space of all sequences.