standard basis

in $\text{[math]}$ , let $\text{[math]}$ ; $\text{[math]}$ is readily seen to be a basis for $\text{[math]}$ and is called the standard basis for $\text{[math]}$ .
in $\text{[math]}$ , the set $\text{[math]}$ is a basis. we call this basis the standard basis for $\text{[math]}$ .
[cite:;also in @calc_hubbard_2015 definition 1.1.8]

for the vector space $\text{[math]}$ , we call $\text{[math]}$ the standard ordered basis for $\text{[math]}$ . simiarly for the vector space $\text{[math]}$ , we call $\text{[math]}$ the standard ordered basis for $\text{[math]}$ .
[cite:;taken from @algebra_insel_2019 chapter 2.2 the matrix representation of a linear transformation]

consider the set $\text{[math]}$ which is a standard basis of the vector space $\text{[math]}$
also the dimension of this set is 2

when put in a matrix, a standard basis forms an identity matrix

consider the set $\text{[math]}$ which is a standard basis of the vector space $\text{[math]}$
we put the vectors in a matrix:
$\text{[math]}$ which is an identity matrix of the 3rd degree
$\text{[math]}$

when we multiply a matrix $\text{[math]}$ by the $\text{[math]}$ th vector of the standard basis the result is the $\text{[math]}$ th column of $\text{[math]}$ . i.e., $\text{[math]}$ .