information:filtering, structural:expansion, linguistic:translation
Natural basis vectors

Let V = R n {\displaystyle V=\mathbb {R} ^{n}} and consider the following elements in V {\displaystyle V}, known as the natural basis vectors:

Then e 1, e 2, …, e n {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\ldots,\mathbf {e} _{n}} are linearly independent.

Suppose that a 1, a 2, …, a n {\displaystyle a_{1},a_{2},\ldots,a_{n}} are real numbers such that

Since

then a i = 0 {\displaystyle a_{i}=0} for all i = 1, …, n. {\displaystyle i=1,\ldots,n.}