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- There is a cubic polynomial of the form Ax
3
2
- C which contains the four points (3, 47), (2, 10),
(− 1 , −5) and (− 2 , −38).
1A. Set up the system of equations you need to solve in order to find A, B and C.
1B. Solve the system, and find the polynomial. This is not a least squares problem!
- Suppose W is the column space of A =
and let z =
a
b
c
d
3A. What equation(s) does z have to satisfy in order for z to be in W
⊥ ?
3B. What matrix equation represents the answer to (3A)?
3C. Use (3B) to find a basis for W
⊥ .
3D. Let S be the set of those two column vectors in A. Note that S is orthogonal. Produce an
orthonormal set of vectors which also spans W.
3E. Use S and our dot “product formula” to find the projection w of y =
onto W.
3F. What is the distance from w to y?
- Let A =
; then^ A^ is diagonalizable. Find^ P^ and^ D^ with the “right properties”.
Facts: One eigenvalue for A is 2 and one eigenvector for A is
Show all your steps!
- Suppose that S = {u 1 , u 2 ,... , up} is a set of vectors all having length 2, and that W =span(S).
Suppose furthermore that S ∪ {b} = {u 1 , u 2 ,... , up, b} is orthogonal, and b is not equal to any of the u’s.
6A. Show that b is perpendicular to any vector in W , and hence is in W
⊥ .
6B. Is S a basis for W? Explain.
6C. Can b be a linear combination of the members of S? Explain any possiblilities.
- Let G =
and let b =
b 1
b 2
b 3
b 4
7A. Is every such b in the column space of G? If not, describe any conditions that must be met.
7B. What is the dimension of the column space of G?
7C. Find all solutions of Gx =
. Express your answer in terms of a particular solution and the
solutions of the corresponding homogeneous equation.
7D. What is the dimension of the null space of G?
7E. Find a basis for the null space of G.
7F. Find a basis for the row space of G.
- Suppose D =
and b =
; then the least squares solution to Dx = b is
(^) + x 3
(^) where x 3 is free.
10A. Do the columns of D form orthogonal set? Explain.
10B. Find the projection of b onto the column space of D. Show your work.
