Examples from Coordinates and bases

Eight vectors in $\mathbb R^5$

Suppose the row-reduced matrix has columns corresponding to eight original vectors:

The leading columns are $1,4,5,7$. Retaining the four original vectors in those positions gives a basis for the span of the eight vectors, so that span has dimension $4$.

Extending a set in $\mathbb R^3$

Let

Writing $u_1,u_2,u_3$ for these vectors, the first two are independent and

so $\{u_1,u_2\}$ is a basis for $\operatorname{span}(S)$. Following the construction, adjoin the standard basis columns:

The leading columns are $1,2,3$. Keeping the corresponding original columns gives the basis

for $\mathbb R^3$.

Coordinates in the standard polynomial basis

For the ordered basis

of $P_n(F)$ over the chosen scalar field $F$, the polynomial

has coordinate vector

Coordinates in a non-standard polynomial basis

Let

First check that these polynomials form a basis. If $\lambda_1(1+x)+\lambda_2(x+x^2)+\lambda_3(1+x^2)=0$, equating the constant, linear and quadratic coefficients gives

These equations force all three coefficients to be zero. Thus $B$ is an independent set of three vectors in the three-dimensional space $P_2(\mathbb R)$, so it is a basis. To find the coordinates of $x^2+x+1$, solve

Equating coefficients gives

Thus

and

An orthonormal basis of $\mathbb R^3$

The set

is orthonormal, hence independent. It has three vectors in $\mathbb R^3$, so it is a basis. For

the orthonormal formula gives