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

Answer:

(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 $\lambda \in \mathbb{R}$

Step-by-step 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 $\lambda\neq 1$ 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.