Download Math 205B&C Final Exam - Parabolas and Linear Equations and more Exams Linear Algebra in PDF only on Docsity!
Useful Information for problem 3
[A|I 5 ] =
AT^ =
1a. The points (1, 11), (2, 10), (4, 14) and (5, 19) do all belong to a parabola of the form β 2 x^2 + β 1 x + β 0. Find β 2 , β 1 and β 0 by setting up and solving an appropriate system of linear equations.
1b. The data points gathered in a lab experiment were supposed to lie on a parabola of the form β 2 x^2 + β 1 x + β 0. The points were (1, 12), (2, 9), (4, 19) and (5, 22). Show why there can be no such parabola.
1c. Find the best fit (least-squares) parabola of the form β 2 x^2 + β 1 x + β 0 for the data points in (1b). Show all your work.
- Suppose that M is a 3 by 3 matrix and
(^) is in its null space.
Suppose also that both
(^) and
(^) satisfy Mx = 4x.
2a. What is the dimension of the eigenspace corresponding to λ = 4? Explain.
2b. What is the characteristic polynomial of M?
2c. Find M (Hint: first find “P and D” as in the diagonalization theorem. Show P and D as part of your work).
- Let A =
; see the useful information on page 0.
Label the six columns of A as a 1 ,... , a 6. In your answers, wherever possible, use the “a 1 ,... , a 6 ” notation instead of explicitly writing out the column vectors. 3a. Find a basis B for Col(A).
3b. Write a 6 as a linear combination of the basis vectors in B.
3c. Let R be the reduced row echelon form (RREF) of A. Is the first column of R in Col(A)? Explain how you know. (Hint: Try to express that first column as a linear combination of the members of B). Does Col(A) = Col(R)? Explain.
3d. Find all solutions of Ax = a 6 expressed in the form p+vh where p is a particular solution of Ax = a 6 and vh is all solutions of the corresponding homogeneous equation Ax = 0. (Note: the information on page 0 should be sufficient; you shouldn’t need to do any additional row reductions)
- Let F be the vector space of continuous functions f : R → R as discussed in class. Let H = {f ∈ F | if a < b then f(a) ≤ f(b)}.
5a. Which of the following functions is in H? (Circle them)
u 1 = ex^ u 2 = x^2 + 1 u 3 = 3
x
5b. Is the zero-vector of F in H? Explain.
5c. Is H closed under vector addition? Show why or give a counterexample.
5d. Is H closed under scalar multiplication? Show why or give a counterexample.
- Suppose that V and W are vector spaces and T : V → W. What does it mean to say T is a linear transformation?
