For example what is the matrix of the linear transformation from R3 ...

Example 6. The data points at the right have been observed.
It is desired to fit a quadratic regression model:
y = x 2 + x + .
Find the least squares quadratic polynomial and plot its graph
along with the data points. Calculate the residual vector .
Solution: We write the six equations, one for each data point as
the vector equation:
0
0
1
1
4
4
0
0
1
1
2
2
X + = y.
1
1
0
1
2
1
1
3 0
1
4 1
1
1
5
1
6
2
The condition that be orthogonal to the columns of X is:
34
18
10
X T = 0
X T (y – X) = 0
X T X = X T y
18
10
6
10
13
6
7 .
6
5
We can solve this by solving the system of equations by elimination
or using technology to calculate the matrix inverse. A good website
is: http://www.bluebit.gr/matrix-calculator/calculate.aspx
The solution is
^
34
^
18
^
10
18
10
6
10
6
6
1
giving us the best fit parabola
The residual vector is
= y – X^ =
13
3
1
7
6
4
5
1
1 2
y = (x – x + 1) .
2
0
0
1
0
0
1
1
1
1
4
2
4
0
0
1
1
2
2
6
13
3
1
1
1
1
1
1
1 1 1
1
1
2 2
1
1
1
1
1
1
As usual, check that is orthogonal to the columns of X .
1 13
1
1
3
7
1
2
2
5
1
x y
0 0
0 1
1 0
1 1
2 1
2 2
23C approximation and best-fit. 6
3
2
1
0
3
2
1
0
y
y
0 1 2 3
0
0
0
0
1
1
u v
1
1
4
2
4
2
u•u = 34
v•v = 10
w•w = 6
u•v = v•u = 18
u•w = w•u = 10
v•w = w•v = 6
1
1
1
w
1
1
1
y= (x^ 2 - x + 1)/2
0 1 2 3
x
x
0
1
0
y
1
1
2
Having seen this picture, can
you see a way you might have
deduced that this would have to
be the answer without doing
any work at all?