How to find the rank of the set of vectors $\{(5,x), x\in \mathbb R\}$, $S \subset \mathbb{R}^2$?

Question

Let $S$ be the set of column vectors such that $\{(5,x), x\in \mathbb R\}$ and $S$ is a subset of $\mathbb{R}^2$.

What will be the rank of $S$ in this case?

Also, what will be the basis and how many basis will we have?

Answer 1

I assume you mean by $rank(S)$ the dimension of $span(S)$.

First of all $$span(S) \subset \mathbb{R}^2 \Rightarrow dim(S) \leq 2$$

Consider $u = \begin{pmatrix}5 \\ 0 \end{pmatrix}$ and $v = \begin{pmatrix}5 \\ 1 \end{pmatrix}$: $$u,v \in S \mbox{ and } rank\begin{pmatrix}5 & 5 \\ 0 & 1 \end{pmatrix} = 2 \Rightarrow rank(S) = 2$$

Concerning possible bases of $span(S)$ belonging to $S$:

Just choose $u = \begin{pmatrix}5 \\ 0 \end{pmatrix}$ and $v_x = \begin{pmatrix}5 \\ x \end{pmatrix}$ with $x \neq 0$.