Assume u4 is not a linear combination of {u1,u2,u3}

Question

Assume u4 is not a linear combination of {u1,u2,u3}

 

A. {u1,u2,u3,u4} is never a linearly dependent set of vectors.
B. {u1,u2,u3,u4} could be a linearly independent or linearly dependent set of vectors depending on the vector space chosen.
C. {u1,u2,u3,u4} is a linearly dependent set precisely when {u1,u2,u3} is a linearly dependent set.
D. {u1,u2,u3,u4} is a linearly independent set of vectors unless one of {u1,u2,u3} is the zero vector.
E. {u1,u2,u3,u4} is always a linearly dependent set of vectors.
F. none of the above

Answer 1

I suggest none of the above (F).  Let us look at each choice in turn.

 

(A) While u₄ is not dependent on u₁, u₂, and u₃, we know nothing about linear dependence among u₁, u₂, and u₃.  Hence, (A) cannot be right.

 

(B) Since u₄ is independent of the other 3, the set of 4 cannot be linearly dependent.  But again, we do not know the relation between u₁, u₂, and u₃.  The set of 4 could be independent if u₁, u₂, and u₃ are independent.  But (B) says that depends on the vector space, which is not correct.  It only depends on the relation between u₁, u₂, and u₃.  So (B) is not entirely correct.

 

(C) is false, because u₄ is included in the set and it is given that u₄ is independent of u₁, u₂, and u₃.

 

(D) is true only if u₁, u₂, and u₃ are independent, which we do not know, so we cannot choose (D).

 

(E) is false because we are given that u₄ is independent of u₁, u₂, and u₃.

 

That leaves us with none of the above.