# Standard Basis Vectors

We refer to the vectors $e_1=(1,0)$ and $e_2=(0,1)$ as the standard basis vectors in $R^2$. A vector $a=(a_1,a_2)$ can be expressed in term of the basis vectors $e_1$ and $e_2$ as

• $a=a_1e_1+a_2e_2$,

since

• $a_1e_1+a_2e_2=a_1(1,0)+a_2(0,1)=(a_1,0)+(0,a_2)=(a_1,a_2)=a$.

as illustrated in:

The standard basis vectors $e_1$ and $e_2$ and a linear combination $a=(a_1,a_2)=a_1e_1+a_2e_2$ of $e_1$ and $e_2$.

We say that $a_1e_1+a_2e_2$ is a linear combination of $e_1$ and $e_2$ with coefficients $a_1$ and $a_2$. Any vector $a=(a_1,a_2)$ in $R^2$ can thus be expressed as a linear combination of the basis vectors $e_1$ and $e_2$ with the coordinates $a_1$ and $a_2$ as coefficients.

Example: We have $(3,7)=3, (1,0)+7, (0,1)=3e_1+7e_2$.