Projective basis

Projective basis:

A projective basis for ${\cal P}^n$ is any set of $n+2$ points no $n+1$ of which are linearly dependent.

Canonical basis:

$$\underbrace{ \left[ \begin{array}{c} 1\\ 0\\ \vdots \\ 0 \end{arr... ...\begin{array}{c} 1\\ 1\\ \vdots \\ 1 \end{array} \right] }_{\mbox{unit point}}$$

Change of basis:

Let ${\bf e_1},{\bf e_2},\ldots,{\bf e_{n+1}},{\bf e_{n+2}}$ be the standard basis and ${\bf a_1},{\bf a_2},\ldots,{\bf a_{n+1}},{\bf a_{n+2}}$ be any other basis. There exists a non-singular transformation $\left[ {\bf T} \right]_{(n+1)\times (n+1)}$ such that:

$${\bf T} {\bf e_i} = \lambda_i {\bf a_i}, \forall i = 1,2.\ldots,n+2$$

${\bf T}$ is unique up to a scale.

Proof:
From the first $n+1$ equations we have that ${\bf T}$ must be of the form

$${\bf T} = \left[ \begin{array}{cccc} \lambda_1 {\bf a_1} & \lambda_2 {\bf a_2} & \ldots & \lambda_{n+1} {\bf a_{n+1}} \end{array} \right]$$

${\bf T}$ is non-singular by the linear independence of ${\bf a}$ 's.

The final equation gives us:

$$\left[ \begin{array}{cccc} \lambda_1 {\bf a_1} & \lambda_2 {\bf a... ...{array}{c} 1\\ 1\\ \vdots\\ 1 \end{array} \right] = \lambda_{n+2}{\bf a}_{n+2}$$

which is equivalent to:

$$\left[ \begin{array}{cccc} {\bf a_1} & {\bf a_2} & \ldots & {\bf ... ...da_2\\ \vdots\\ \lambda_{n+1} \end{array} \right] = \lambda_{n+2}{\bf a}_{n+2}$$

Since the matrix on the left hand side of the above equation is of full rank (by linear independence of ${\bf x}$ 's), the ratios of the $\lambda_i$ are uniquely determined and no $\lambda_i$ is 0.