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:

$\displaystyle \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:

$\displaystyle {\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.

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

$\displaystyle {\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:

$\displaystyle \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:

$\displaystyle \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.
Subhashis Banerjee 2008-01-20