Is this True or False ? Justify please

Question

If X₁,X₂ and X₃ are vectors in R³ such that {X₁,X₂} and {X₁,X₃} are linearly independent sets, then {X₁,X₂,X₃} is a linearly independent set.

Answer 1

Here is a counterexample to show it is false.

 

X₁ = (1, 0, 0)

X₂ = (0, 1, 0)

X₃ = (0, 2, 0)

 

X₁ and X₂ are linearly independent.

X₁ and X₃ are linearly independent.

But X₂ is not linearly independent of X₃, so {X₁,X₂,X₃} is not a linearly independent set.