Download Final Exam in Mathematics 20F: Problems on Linear Algebra and Eigenvalues and more Exams Linear Algebra in PDF only on Docsity!
Name: Time section meets:
Mathematics 20F Professor Linda Rothschild June 13, 1996 Final Exam-Version A
Instructions: Answer all questions. Use calculators for computation whenever it is quicker. There is not enough time to do all problems by hand. Indicate what the calculator has shown. Show some work or reason for each answer. For justification, you can mention any fact cited in the text, but you cannot cite something shown in a homework problem. There are two versions of this exam; anyone who gives some answers corresponding to the other version will receive a 0 for the final.
Total 200
Have a great summer!
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- (10 pts.) Find the general solution of the system
x 1 + 5x 3 + 6x 4 = 6 x 1 + x 2 + 2x 4 = 1 3 x 1 + 2x 2 + 6x 3 + 11x 4 = 11,
- (10 pts.) A linear transformation T : R^4 → R^4 is defined by
T (
x 1 x 2 x 3 x 4
x 1 + 2x 2 + x 3 x 2 − x 3 x 1 + 3x 2 x 3 + x 4
(a) Find the standard matrix for T.
(b) Find a basis for the kernel of T.
- (30 pts.) For each statement, mark it True or False. If true, give a brief reason. If false, explain why or give a “counterexample”. No credit if reason is wrong.
(a) If A and B are 2 × 2 matrices, with A invertible, and if AB = 0, then B = 0.
(b) If A is a 2 × 2 matrix which is diagonalizable, then A is symmetric.
(c) If the eigenvalues of a 3×3 matrix A are 0, 1, and 2, then A is diagonalizable.
(d) If B is diagonalizable and B−^1 exists, then B−^1 is also diagonalizable.
(e) Suppose A and B are square matrices, and B is obtained from A by row operations. Then every eigenvalue of A is an eigenvalue of B.
- (12 pts.) Let V be the plane in R^4 spanned by the vectors
and
Find the vector in V closest to y =
- (30 pts.) For each of the following, read carefully and determine if it is true or false. GIVE ANSWER ONLY, True or False. NO REASONS NEED BE GIVEN, NOR WILL REASONS BE GRADED. EACH PART IS WORTH ±3 POINTS. IF CORRECT, IT WILL BE GRADED +3. IF INCORRECT, IT WILL BE GRADED -3. (However, the minimum score for this problem will be 0.)
(a) If A is any matrix, the system Ax = 0 must have at least one solution.
(b) If A is an n × n orthogonal matrix, then ‖Ax‖ = ‖x‖ for all x ∈ Rn.
(c) If a square matrix A is diagonalizable, then its rows must be linearly inde- pendent.
(d) If V is a vector space and there is no set of n vectors which spans V , then dim(V ) > n.
(e) If there is a linearly DEPENDENT set {v 1 , v 2 , v 3 , v 4 } of vectors in V , then dim(V ) < 4.
(f) The mapping T (
[
x 1 x 2
]
[
x 1 x 2 + 1
]
with a, b ∈ R is a linear mapping of R^2
onto R^2.
(g) If A and B are square matrices which are similar to each other, then they must have the same eigenvalues.
(h) If A and B are square matrices which are similar to each other, then they must have the same eigenvectors.
(i) If A is a square matrix, and 0 is an eigenvalue of the matrix A − 2 I, then 2 must be an eigenvalue of A.
(j) There exists a linear mapping of R^3 onto R^4.
- (7 pts.) Give an example of 2 × 2 matrices A and B such that A and B have the same characteristic poynomial, but A is diagonalizable while B is not diagonalizable. (Not necessary to show why your example works.)
(d) If A is not invertible, find det A.
(e) If A^2 = 0, show that A is not invertible.
- (12 pts.) If {v 1 , v 2 , v 3 } is a linearly independent set of vectors in a vector space V and v 4 is a vector in V which is not in the span of {v 1 , v 2 , v 3 }, show (carefully) that {v 1 , v 2 , v 3 , v 4 } is linearly independent.
- (12 pts.) Describe the set of all polynomials p(t) = a + bt + ct^2 + dt^3 whose graph passes through the points (0, 1), (1, −2), (2, 0).
