Let u4 be a vector that is not a linear combination of {u1,u2,u3,}.

Question

Select the best statement.

A) span{u1, u2, u3} = span{u1, u2, u3, u4}

B) There is no obvious relationship between span{u1, u2, u3} and span{u1, u2, u3, u4}

C) span{u1, u2, u3} is a proper subset of span{u1, u2, u3, u4}

D) We only know that span{u1, u2, u3} ⊆ span{u1, u2, u3, u4}

E) span{u1, u2, u3} = span{u1, u2, u3, u4} when none of {u1, u2, u3, } is a linear combination of the others.

F) We only know that span{u1, u2, u3, u4} = span{u1, u2, u3}

G) None of the above.

I have tried B and G thinking that there is not an explicit relationship between the two sets, however, both those choices are incorrect.

Answer 1

C is the answer.

Reason: Every linear combination of u1,u2,and u3 lies inside the sapn{u1,u2,u3,u4}. But no linear combination containing u4 lies inside the span{u1,u2,u3}.