Question

True or False, and WHY!!! If V1........V4 are in R4, and V3=2V1+V2, then {V1,V2,V3,V4} is linearly dependent. I know that V3 is a linear combo of V2 and V1, but doesnt the theorem state that in order for the set to be dependent, one vector hass to be a linear combo or multiple of ALL the other vectors, not just SOME of the other vectors. Please explain in detail.

Answer 1

The set {V1, V2, V3, V4} is linearly dependent because V3 can be expressed as a linear combination of V1 and V2.

Step-by-step explanation:

The statement that if V1, V2, V3, and V4 are vectors in R4, and V3 = 2V1 + V2, then the set {V1, V2, V3, V4} is linearly dependent is True. This is because one vector (V3) is expressible as a linear combination of two others (V1 and V2), which is enough to establish linear dependence. Linear dependence means that there exists at least one non-trivial combination of the vectors that equals the zero vector. In this case, that combination is -2V1 - V2 + V3 + 0V4 = 0. Furthermore, it is not required that one vector must be a linear combination of all other vectors; it is sufficient if one vector can be written as a linear combination of any other vectors in the set.