Final answer:
It is false to assume that adding a fourth vector u₄ to a linearly independent set of vectors {u₁, u₂, u₃} will always result in a new set that is also linearly independent.
Step-by-step explanation:
B. False. The addition of u4 to a set of linearly independent vectors does not guarantee that the new set will also be linearly independent.
In linear algebra, a set of vectors is considered linearly independent if none of the vectors can be expressed as a linear combination of the others. Suppose you have a set of vectors {u1, u2, u3} that is linearly independent. This means that there are no scalars a, b, and c, not all zero, such that au1 + bu2 + cu3 = 0. When a fourth vector u4 is added to the set, the set {u1, u2, u3, u4} could still be linearly dependent if u4 can be expressed as a linear combination of u1, u2, and u3. Therefore, the original statement is false; the inclusion of an additional vector to an independent set does not necessarily preserve the set's linear independence.