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

Example 8. At the right is tabulated the mid-term test mark (out of
10) and the final marks for four of Jason’s friends who took the course
last year. Jason’s mid-year test mark is 7. He decides to use a linear
model
y = mx + b
to predict his final mark. What does he discover?
Solution.
We use a “least squares” model. The equations are:
90 = 8m + b
85 = 7m + b
95 = 9m + b
75 = 7m + b
In matrix form:
8
7
9
7
Xb = y
1
90
1
m
85
.
1
b
95
1
75
The least squares solution b^ is the solution of the equation:
243
31
m^
243
b^
31
The best fit recursive equation is
(X T X)b^ = X T y.
31m^
2695
4
b^
345
31
4
1
2695
345
y = (85x + 290)/11
1 85
.
11 290
Jason’s estimate of his final mark is y = (85×7 + 290)/11 = 80.4.
student Mid Final
Brenda 8 90
Alnoor 7 87
Kit 9 94
Tim 7 75
23C approximation and best-fit. 8