Answer to Question #141083 in Linear Algebra for Julia

Question

It is possible for a set S of three vectors from R^3 to be linearly dependent, even though every subset of two vectors from LaTeX: S\:Sis linearly independent

Answer 1

It is true. Take the set S to be $\{e_1,e_2,e_1+e_2\}$ . Linear dependency is clear. $e_1+e_2+(-1)(e_1+e_2) =0.$ Now $e_1,e_2$ are linearly independent. Since $a_1e_1+a_2+e_2=(a_1,a_2,0)$ . Hence sum is zero implies , $a_1=a_2=0.$ Now similarly $e_1,e_1+e_2$ and $e_2,e_1+e_2 ​$ are linearly independent. Hence done.