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MATH 320 PRACTICE MIDTERM 2
This practice midterm is about twice the length that the actual midterm will be. You are encouraged to take it in exam conditionsāno notes, calculator, book, etc., and keeping time constraints in mind. Solutions will be posted on Tuesday, Nov. 6. For all problems, you must fully explain your reasoning. Write in complete sentences when appropriate.
Problem 1. Let f : R^4 ā R^3 be the homomorphism whose associated matrix with respect to the standard bases is given by ļ£« ļ£
and let g : R^3 ā R^4 be the homomorphism whose associated matrix with respect to the standard bases is given by ļ£« ļ£¬ ļ£¬ ļ£
(1) Determine, with justification, the matrix corresponding to the homomor- phism g ā¦ f with respect to the standard bases. (2) Is the matrix you found in (1) equal to the matrix corresponding to the homomorphism f ā¦ g?
Problem 2. Consider the homomorphism f : R^3 ā R^3 corresponding to the matrix
A =
with respect to the standard bases.
(1) Verify that f is an isomorphism, and compute the matrix Aā^1. (2) Given any vector
~b =
b 1 b 2 b 3
ļ£ø ā R^3 ,
what is the solution ~x to the matrix equation A~x = ~b?
Problem 3. (1) Why is there a homomorphism f : R^3 ā R^3 uniquely speci- fied by the requirements
f
ļ£ø (^) f
ļ£ø (^) f
(2) Determine the matrix RepE 3 ,E 3 (f ) of f with respect to the standard bases.
(3) Show that f is an isomorphism. (4) Determine the matrix of f ā^1 with respect to the standard bases.
Problem 4. Let f : R^3 ā R^3 be the homomorphism given by
f
x y z
x + y + 3z x + 2y + 5z y + 2z
(1) Represent f as a matrix with respect to the standard bases. (2) Determine bases for the image (i.e. range space) of f and the kernel (i.e. null space) of f. (3) What are the rank and nullity of f? (4) Determine bases for the orthogonal complements (im f )ā„^ and (ker f )ā„.
Problem 5. Consider the homomorphism f : R^2 ā R^3 given by
f
x y
x ā 2 y 3 y x + 2y
(1) Is f injective? Surjective? Bijective? (2) Does f admit a left inverse? If so, find one, and represent it as a matrix with respect to the standard bases. (3) Does f admit a right inverse? If so, find one, and represent it as a matrix with respect to the standard bases.
Problem 6. Let f : P 5 ā R^3 be the homomorphism given by
f (p(x)) =
p(1) p(2) p(3)
where P 5 is the space of polynomials of degree at most 5.
(1) Determine a basis for the null space of f , and compute the nullity of f. (Hint: weāve already done this in class and on homework, just in a different language. Donāt write too much). (2) What is the rank of f?
Problem 7. Which of the following functions are homomorphisms? Justify your answers.
(1) f : Pn ā Pn+1 given by f (p(x)) = xp(x). (2) f : Pn ā Pn given by f (p(x)) = p(x) + pā²(x), where pā²^ is the derivative. (3) f : Pn ā Pn given by f (p(x)) = pā²(1)p(x).
Problem 8. Fix an angle Īø. Let f : R^3 ā R^2 be the function given by projecting a vector ~v ā R^3 vertically to the xy-plane, then rotating it in that plane by an angle of Īø.
(1) Is f a homomorphism? Justify your answer. (2) Explicitly describe the preimage f ā^1 ({~ 0 }). (3) Explicitly describe the preimage f ā^1 ({~e 1 }).
Problem 9. Let f : Rn^ ā Rm^ be a homomorphism. Let N be the null space of f and let R be the image of f. Consider the four spaces N, N ā„, R, Rā„. There are six pairs of two of these spaces, namely (N, N ā„), (N, R), (N, Rā„), (N ā„, R), (N ā„, Rā„),
