1. ## Linearly dependent

What does linearly dependent mean? My teacher and textbook do not teach nor define it. What is it? An example of one is found below, however, how does one solve it?

17. Determine if the following vectors are linearly dependent.
u = [4, 1, 0], v = [-8, 5, 2,], w = [0, 7, 2].

How does one solve this and prove that it is linearly dependent?

2. Originally Posted by Barthayn
What does linearly dependent mean? My teacher and textbook do not teach nor define it. What is it? An example of one is found below, however, how does one solve it?

17. Determine if the following vectors are linearly dependent.
u = [4, 1, 0], v = [-8, 5, 2,], w = [0, 7, 2].

How does one solve this and prove that it is linearly dependent?
A set of vectors $\{\mathbf{u},\,\mathbf{v},\,\mathbf{w}\}$ is said to be linearly dependent if $\exists\,\alpha,\,\beta,\,\gamma$ not all zero such that $\alpha\mathbf{u}+\beta\mathbf{v}+\gamma\mathbf{w}= \mathbf{0}$.

So in your question, do there exists constants $\alpha,\,\beta,\,\gamma$ not all zero such that $\alpha[4,1,0]+\beta[-8,5,2] +\gamma[0,7,2]=[0,0,0]$?

Spoiler:
There is a linear dependence, but I leave it for you to find what constants make it linearly dependent.

3. Not to be antagonistic, but I find it difficult to believe that a textbook on Matrix Theory/Linear Algebra/anything to do with vectors wouldn't define the concept of linear (in)dependence.