Show that this set of vectors with sum of coordinates equals $2$ is linearly independent

Question

Let

\begin{equation*} A = \{(x_1,x_2,\dots,x_n) \in \mathbb{R}^n :\, x_1+x_2+\cdots+x_n = 2\} \end{equation*}

be a set of vectors in $\mathbb{R}^n$. What is the rank of $A$?

My attempt : I conjecture that the rank is $n$. That is why I try to show that they are linearly independent:

Say $a_1x_1 + \cdots +a_n x_n = 0$, I'm trying to show that this holds iff $a_i =0$.

Any help would be appreciated. Thanks.

Answer 1

If the rank of a set of vectors is the dimension of the vector space that they span, then the answer is $n$, because$$A\supset\bigl\{(2,0,0,\ldots,0),(0,2,0,\ldots,0),\ldots,(0,0,0,\ldots,0,2)\bigr\}.$$