Question regarding sets of vectors and linear combinations. I stumbled on this passus in a material that I am reading to better understand linear independence. (the material can be found here: < * * * The set $e_1, e_2, e_3, u = (2, 0, 1)$ is a spanning set of $\mathbb{R}3$, Why? _Can someone please explain what the line above actually means?_ Here the vector $\vec{v} = (1, 0, 1)$ can be written as a linear combination of these $4$ vectors in at least two dierent ways: $\vec{v} = 1e_1 + 0e_2 + 1e_3 + 0u$ $\vec{v} = (-1)e_1 + 0e_2 + 0e_3 + 1u$ _The first definition of the vector v I can understand, but I don't understand how the second definition works..._ Thank you for your help!

It means that every vector in $\mathbb R^3$ can be written as a linear combination of $e_1 , e_2 , e_3$ and $u$, where $u = (2,0,1)$. You have written the second equation incorrectly. It should be written:

$$\vec{v} = (-1)e_1 + 0e_2 + 0e_3 + u$$

You can verify that this is correct as well, and so there is no unique representation of $v$ in terms of $\\{ e_1,e_2,e_3,u \\}$ (you will learn that this means that these vectors are _linearly dependent_ ).