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1a) The four points (1, 9) (3, −37) (4, 10) and (7, −19) are definitely not colinear. Find the least- squares line for these points, expressed in the form y = β 0 + β 1 x. Show all your work, from the design-matrix equation through the solution of the least-squares problem.
1b) For your answer in 1a, what are the corresponding four expected values corresponding to 1,3, 4 and 7 respectively?
1c) Find the least squares error (the sum-of-squares of the residuals) for the line in (1a)
1d) Find the equation of the line containing (1, 9) and (7, −19). What is the sum of squares of the residuals for this line?
2a) The three points (1,6) (4,3) (5,6) do lie on a parabola of the form y = β 0 + β 1 x + β 2 x^2. Set up the system of equations you need to solve in order to find β 0 , β 1 and β 2 , and solve it.
- Suppose T : R^2 → R^3 satisfies T (
([
])
(^) and T (
([
])
3a) Find T (
([
])
3b) Find T (
([
])
- Let B =
5a) Do the columns of B form an orthogonal set? Explain.
5b) Find the projection, bproj of b =
onto col(B).
5c) Find the vector z satisfying b = bproj + z.
- Let M =
[
]
6a) Find the inverse of M using simultaneous row reduction on M and I 2. I want to see the steps; that’s the important issue here. Avoid fractions if possible.
6b) What’s the (formula for the) inverse of
[
a b c d
]
, in general? And when does it exist?
6c) Does the formula in (6b) give you the same answer as you got in (6a)?
- Let B =
8a) Find det(B). Show your work.
8b) Find the eigenvalues of B. Show your work.
8c) Find det(B−^1 ).
8d) Find det(B^3 ).
