True or False if u1 is not in the span of u2 u3 then u1 u2 u

True or False: if u1 is not in the span of {u2, u3}, then {u1, u2, u3} is linearly independent.

Please explain

Solution

True

Clearly {u₂ , u₃ } is independent. If they are dependent then anyone of them can be expressed as a multiple ( a constant times) of the other ., So that span of  {u₂ , u₃ } iseither span {u₂} or span {u₃} . Since that is not the case {u₂ , u₃ } is independent.

Now since u₁ cannot be expressed asa linear combination of u₂ , u₃ , the set {u₁ , u₂ , u_{3} is independent}