I am doing an exercice where I have to put the normal equation of a least squares problem, in the form: $C\alpha = d$
The given points are the following: $(-2,2), (2,4), (3,6)$ And the approximation function is this one:
$$f(x) = \alpha_1 x + \alpha_2 x^2$$
My solution is the following one:
$$ \left[ \begin{array}{cc|c} 17&27&22\\ 27&113&70 \end{array} \right] $$ with $C$ the $(2x2)$ matrix and $d = $${22}\choose{70}$$ t$ (sorry for my weak $\LaTeX$ knowlegde)
So first question: Is my solution right?
Second, it is asked after that to tell WITHOUT CALCULATIONS how the error of the solution would change, if we change the approximation function to:
$$g(x) = \alpha_1x + \alpha_2x^2 + \alpha_3x^4$$
How to do that without calculation? What would be a right answer to this?
PS: I would have calculated again the $C\alpha=d$ and checked the condition number of the new $C$ and compared it to the condition number of the previous $C$ and would have said that the new one is ill-conditioned. But we have to answer without calculations.
On your first question: Almost, but I get $(-2)^2\cdot2+2^2\cdot4+3^2\cdot6=78$ where you have $70$.
On your second question: You now have three functions to approximate three values. You can readily check that the three vectors of their values at the three points $2$, $-2$, $3$ are linearly independent: $(-2,2,3)$, $(4,4,9)$ and $(16,16,81)$. Thus the approximation error will be zero.