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MATH 320 FINAL
For all problems except Problem 9, you must fully explain your reasoning. You may use any facts proved in class. Write in complete sentences when appropriate. No notes, calculator, book, etc., are allowed.
Problem 1 (20 points). (1) Determine all solutions of each of the three following systems of equations.
−x + 3y + 2z = 1 −x + 3y + 2z = 0 −x + 3y + 2z = 0 y + 4z = 0 y + 4z = 1 y + 4z = 0 −z = 0 −z = 0 −z = 1. (2) Let f : R^3 → R^3 be the homomorphism
f
x y z
−x + 3y + 2z y + 4z −z
Represent f as a matrix with respect to the standard bases. (3) Show that f is an isomorphism. (You may assume f is a homomorphism.) (4) Represent f −^1 as a matrix with respect to the standard bases.
Problem 2 (15 points). Let
A =
B^ =
(1) Compute det(AB). (2) Compute det(A + B). (3) Let f : R^4 → R^4 be the homomorphism f (~v) = (A + B)~v. Is f injective? Surjective? Bijective?
Problem 3 (25 points). Let
A =
(1) Compute the characteristic polynomial pA(x). (2) Compute the eigenvalues of A, and compute their algebraic multiplicities. (3) For each eigenvalue of A, compute a basis of the corresponding eigenspace and compute the geometric multiplicity. (4) Determine if A is diagonalizable. If it is, write an equation of the form D = C−^1 AC, where D is diagonal.
Problem 4 (20 points). (1) Show that there is a unique homomorphism f : R^3 → R^3 satisfying
f
(^) f
(^) f
(^) f
(2) Represent f as a matrix with respect to the standard bases.
Problem 5 (20 points). Let f : V → V be a homomorphism, where V is a vector space. Suppose there are nonzero vectors ~v 1 , ~v 2 ∈ V such that
f (~v 1 ) = ~v 1 and f (~v 2 ) = 2~v 2.
Show that {~v 1 , ~v 2 } is a linearly independent set.
1
Problem 6 (30 points). Define a function f : P 2 → P 2 on the space of polynomials of degree at most 2 by the rule f (p(x)) = p(2x + 1). (1) Show that f is a homomorphism. (2) Compute the representation RepB,B(f ) of f as a matrix with respect to the standard basis B = { 1 , x, x^2 } of P 2. (3) Determine if there exists a basis D of P 2 such that the matrix RepD,D(f ) is diagonal. If there does, find such a basis D, and compute RepD,D (f ).
Problem 7 (30 points). Let
A =
Define a homomorphism f : P 2 → Mat 2 × 2 from the space of polynomials of degree at most 2 to the space of 2 × 2 matrices by the rule f (p(x)) = p(A).
Explicitly, if p(x) = a 2 x^2 + a 1 x + a 0 , then
f (p(x)) = a 2 A^2 + a 1 A + a 0 I. (1) Compute the representation RepB,D (f ) of f as a matrix with respect to the standard bases
B = { 1 , x, x^2 }, D =
of P 2 and Mat 2 × 2. (2) Determine bases for the image (i.e. range space) and kernel (i.e. null space) of f. (3) What are the rank and nullity of f? Is f injective? Surjective? An isomorphism? (4) Compute the minimal polynomial mA(x) of A.
Problem 8 (20 points). Let f : V → W be a homomorphism of vector spaces.
(1) Give a definition of the kernel (i.e. null space) of f. (2) Show the kernel of f is a subspace of V.
Problem 9 (20 points). Let
A =
a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25 a 31 a 32 a 33 a 34 a 35
be a 3 × 5 matrix of real numbers, and let f be the homomorphism f (~v) = A~v. For each of the following statements, say if it is always True (T), always False (F), or Depends (D) on the particular numbers in the matrix. No justifications are required.
(1) The homogeneous system A~v = 0 has infinitely many solutions. (2) For any vector ~b ∈ R^3 , the system A~v = ~b has infinitely many solutions. (3) For any vector ~b ∈ R^3 , if the system A~v = ~b has a solution, then it has infinitely many solutions. (4) f is injective. (5) f is surjective. (6) The row space of A is R^5. (7) The columns of A are linearly independent. (8) The nullity of f is at least 2. (9) The rank of f is 4. (10) We have rank(f ) + null(f ) = 3.
Extra Credit. (Do not attempt this problem unless you are confident in your answers on the rest of the exam.) Let (x 1 , y 1 ),... , (x 5 , y 5 ) ∈ R^2 be five distinct points in the real plane. Let P 2 , 2 be the vector space of polynomials p(x, y) of degree at most 2 in two variables x, y. Use linear algebra to show that some nonzero p(x, y) ∈ P 2 , 2 vanishes at all five points (x 1 , y 1 ),... , (x 5 , y 5 ). (Hint: construct a useful homomorphism f : P 2 , 2 → R^5 .)
2
