if. two vectors are linearly dependent are they multiples

2 min read 21-01-2025

if. two vectors are linearly dependent are they multiples

Yes, if two vectors are linearly dependent, then one is a scalar multiple of the other. This is a fundamental concept in linear algebra. Let's explore why.

Understanding Linear Dependence

Two vectors, u and v, are linearly dependent if there exist scalars a and b, not both zero, such that:

au + bv = 0

This equation signifies that one vector can be expressed as a linear combination of the other. If they were independent, the only solution would be a = b = 0.

The Scalar Multiple Relationship

Let's assume that a is not zero in the equation above. We can rearrange the equation:

au = -bv

Dividing both sides by a (since a is not zero):

u = (-b/a)v

Let's call (-b/a) = k, where k is a scalar. Then:

u = kv

This clearly shows that u is a scalar multiple of v. The same logic applies if we assume b is not zero; we would then find that v is a scalar multiple of u.

Geometric Interpretation

Geometrically, linear dependence means the vectors lie on the same line (or are collinear). One vector is simply a scaled version—a stretch or shrink—of the other. If they point in opposite directions, the scalar will be negative. If they're identical, the scalar is 1. If one is the zero vector, the scalar is zero.

Example

Consider vectors u = (2, 4) and v = (1, 2). We can see that u = 2v. Therefore, they are linearly dependent, and u is a scalar multiple of v.

Exception: The Zero Vector

A special case arises when one of the vectors is the zero vector, 0. Any vector is linearly dependent with the zero vector, as au + 0v = 0 always has a non-trivial solution (a=0, b can be any number). In this instance, it's still true to say the non-zero vector is a multiple (0 times) of the zero vector.

Conclusion

In summary, if two vectors are linearly dependent, one vector is a scalar multiple of the other. This fundamental relationship is crucial in understanding many concepts in linear algebra and has important geometric interpretations. The only exception is when one of the vectors is the zero vector.