# Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent

## Problem 652

Prove that any set of vectors which contains the zero vector is linearly dependent.

## Solution.

Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set.

Then we have the non-trivial linear combination
$1 \cdot \mathbf{0} + 0 \mathbf{v}_1 + 0 \mathbf{v}_2 + \cdots + 0 \mathbf{v}_k = \mathbf{0}.$ This is a non-trivial linear combination because one of the coefficients is non-zero.

Thus by definition, the set $\{ \mathbf{0} , \mathbf{v}_1 , \cdots , \mathbf{v}_k \}$ is linearly dependent.

(a) Find a function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ such that \$g(0) =...