Proving Linear Dependency of A based of $ (SpA)^\perp = \{(a,a-b,b-c,a)\mid a,b,c \in R\} $

Question

I have this question for homework:

Let A be a set of 2 vectors in $R^4$

Given that: $ (SpA)^\perp = \{ (a,a-b,b-c,a)\mid a,b,c \in R \} $

Prove that A is Linearly Dependent.

I think I need to use the facet that the orthogonal complement is equal to the solutions of the linear system $Ax=0$. However I am not sure how (maybe it's not the way at all).

Thanks for the help.

Answer 1

We know that

$\dim (SpA)+\dim((SpA)^\perp)=\dim(\mathbb{R^4})=4$

and $\dim((SpA)^\perp)=3 $ so $\dim (SpA)=1$.

if the 2 vectors in A are not linearly dependent so $\dim((SpA)^\perp)=2 $. so we got that the vectors in A are linearly dependent.