Unique representation of a vector

Question

In a book I am reading the author states without proof that in an $n$-dimensional vector space $X$, the representation of any $x$ as a linear combination of a given basis $e_{1},e_{2},...,e_{n}$ is unique. How to proof that?

Answer 1

A basis is usually defined as a linearly independent, generating set.

Since it is a generating set, by definition, every vector can be expressed as a linear combination.

Suppose some vector has two representations:

$$x=a_1 e_1 +\dots + a_ne_n= b_1 e_1 + \dots + b_n e_n.$$ Then it follows that $$0= (a_1 -b_1)e_1 + \dots + (a_n-b_n)e_n.$$ By the definition of linear independence it follows that $(a_1 -b_1) = \dots = (a_n-b_n)= 0$, which means $a_i = b_i $ for all $i$, that is, the representation is unique.

Answer 2

If $\sum_{k=1}^nc_ke_k=\sum_{k=1}^nc_k'e_k,$ with the $c_k$ and $c_k'$ scalars, then $\sum_{k=1}^n(c_k-c_k')e_k=0$ so... (to continue use the fact that the $e_k$ are linearly independent)