Vector basis definition

Question

Good day all, I;m working throught the book "All the mathematics you missed". When defining a vector basis (definition 1.4.1. pg6), the author states:

A set of vectors $(v_1, ..., v_n)$ form a basis for the vector space V if given any vector v in V, there are unique scalars $(a_1, ... ,a_n)\in\mathbb{R}$ with $(v = a_l v_1 + ... + a_n v_n)$.

What exactly is meant by unique scalars and what is the significance thereof?

Aren't multiple values for $a_1$ and situations where $a_1 = a_i$? Am I missing something?

Answer 1

They are unique in the sense that if you take some other set of scalars $\{b_1, \dots, b_n\}$ and $u = b_1v_1 + \dots + b_nv_n$ then $u \neq v$ unless $a_1 = b_1, \dots, a_n = b_n$.

Answer 2

He means that each $a_i$ is unique, not that they are different between themselves (e.g. he's not claiming $a_1 \not = a_2$). To make an example in $\mathbb{R}^2$, the vectors $v_1 = (1, 0)$ and $v_2 = (0, 1)$ form a basis. The vector $(1,1)$ can only be written as $1v_1 + 1v_2$. There is no other combination $bv_1 + cv_2 = (1,1)$ . That is what is meant by 'unique'. The only possible value of $b$ is $1$, and the only possible value of $c$ is $1$.