This exam contains 16 questions, but you are only required to complete 4 of them. You may choose any four that you like, but clear work must be shown to receive full credit.
Question 1. (25 points) Suppose is a matrix with characteristic polynomial
(a) (5 points) Find all of ‘s eigenvalues and all of the associated algebraic multiplicities.
(b) (5 points) Explain how it is you know that .
(c ) (5 points) Compute the trace of .
(d) (5 points) Suppose is symmetric. What can you conclude about the geometric multiplicities of the eigenvalues? Write down a diagonal matrix that represents .
(e) (5 points) Is it possible that any of the geometric multiplicities for ‘s eigenvalues equals 5? Is it possible that any of them equal 3? Explain your answers.
Question 2. (20 points) Let where and are real vector spaces with dimensions .
Write down as many statements as you can that are logically equivalent to the statement
(Note: you will be awarded 2 points per correct statement, and some bonus points are possible.)
Question 3. (20 points) Does the set of polynomials
form a basis for the vector space ? Prove your answer.
Question 4. (25 points) Let be a matrix. A generalized eigenvector of rank with eigenvalue for the matrix is defined to be a non-zero vector that is in the nullspace of (but is not in the nullspace of . That is
(a) (3 points) Prove that if is a generalized eigenvector of rank (with eigenvalue ) for the matrix , then is an eigenvector for with eigenvalue .
(b) (10 points) Consider the matrix
whose characteristic polynomial is . Hence is an eigenvalue with algebraic multiplicity , but one can check that the geometric multiplicity is . It also follows that is an eigenvalue with algebraic and geometric multiplicities equal to .
Verify that is an eigenvector with eigenvalue for and that is an eigenvector with eigenvalue .
Also verify that is a generalized eigenvector with eigenvalue for .
Finally, represent the matrix using the basis
(c ) (12 points) Suppose is a matrix and that the basis has the following properties:
Explain why these properties imply that and are generalized eigenvectors (both of rank 2) with eigenvalues and , respectively.
Lastly, represent the matrix using the basis
Question 5. (25 points) Consider the linear system of differential equations
(a) (5 points) Describe the behavior of solutions in the phase plane assuming and .
(b) (5 points) Describe the behavior of solutions assuming and .
(c ) (15 points) Describe the behavior of solutions assuming and .
Question 6. (25 points) For this problem is a vector space with an inner product .
(a) (3 points) Using only the definitions of inner product and vector space, prove that is orthogonal to every vector in .
(b) (7 points) The orthogonal complement of a subspace is defined to be
Prove that the orthogonal complement of a subspace is itself a subspace of .
(c ) (15 points) Suppose — where each and are equipped with their standard inner products. Prove that
Question 7. (25 points)
(a) (10 points) Define a relation on the set of all matrices by
Prove that is an equivalence relation.
(b) (8 points) Prove that if , then for every natural number it follows that .
(c ) (7 points) Suppose is a matrix with eigenvalues, associated algebraic multiplicities and associated geometric multiplicities given by
Is there a diagonal matrix that satisfies ? Explain your answer.
Question 8 (25 points)
In a previous homework problem / exam question it was shown that the transpose of a matrix could be viewed as a linear transformation on . For this problem, you do not have to re-prove this fact, and we will use the notation for this linear transformation.
(a) (10 points) Compute the matrix representation
where is the standard basis for the space .
(b) (15 points) Find a basis for that diagonalizes or explain why no such basis exists.
(c ) (Bonus) Find a basis for that diagonalizes or explain why no such basis exists.
Question 9 (20 points)
(a) (6 points) Write down a matrix that cannot be diagonalized, or explain why no such matrix exists.
(b) (8 points) Write down a nilpotent matrix that has two distinct eigenvalues, or explain why no such matrix exists.
(c ) (6 points) Suppose is a matrix with two distinct, real eigenvalues and whose trace . Is it possible that ? Explain your answer.
Question 10 (25 points) Suppose is a matrix that satisfies (in other words is idempotent.)
(a) (7 points) Prove that each eigenvalue of equals or .
(b) (18 points) Prove that is diagonalizable.
Question 11 (25 points) Consider the linear transformation from to (the space of all polynomials)
(a) (10 points) Compute the nullity of by finding a basis for .
(b) (5 points) Find a polynomial that fails to be in the range space
(c ) (10 points) Find all eigenvectors and eigenvalues of .
Question 12 (25 points) Suppose is a square matrix with characteristic polynomial
(a) (2 points) Explain why has eigenvalues and .
(b) (3 points) What is the size of matrix ?
(c ) (5 points) Suppose is symmetric. Compute the geometric multiplicities of ‘s eigenvalues, and explain your answer.
(d) (5 points) Compute the determinant of .
(e) (5 points) Write down the Jordan-Canonical form of assuming and .
(f) (5 points) Suppose there exist two non-zero vectors and that satisfy and . Explain why and are multiples of one another.
Question 13 (25 points) For parts (a) and (b) of this problem let be a vector space with inner product and let be a basis for .
(a) (10 points) Consider the set of vectors that are defined by the (recursive) formulas
Prove that the set is an orthogonal set of vectors. (Suggestion: you may first want to check that and are orthogonal, and then that is orthogonal to both and .)
(b) (5 points) Explain why the orthogonal set from part (a) is still a basis for . Lastly, explain why the new set of vectors, where
is an orthonormal basis for .
(c ) (10 points) Apply the algorithm / recursive formula(s) from parts (a) and (b) to replace the basis for with an orthonormal basis. For this part we have equipped with the inner product
Question 14. (25 points) This question is about one of my other favorite matrices. Here it is:
(a) (2 points) Write a one sentence explanation justifying the claim that is diagonalizable.
(b) (3 points) Here’s why is one of my favorite matrices: you can use it to build a cool sequence of numbers, .
How is this done? You repeatedly multiply against . In particular, we’ll get this sequence going by declaring the first two numbers: and . To get the next number in this list, , you use the given matrix according to the following rule:
This rule can be repeated to generate all the numbers in this list:
Now for the actual question: what crazy famous sequence of numbers does this process produce?
(c ) (10 points) Find a change-of-basis matrix so that
and be sure to find the eigenvalues and , too.
(c ) (10 points) Use your work in part (c ) to find a formula for the -th number in this (famous) sequence, a formula that only uses the eigenvalues and .
Question 15. (25 points) Alex and Kiko play the following game: they start with an empty 2008×2008 matrix and take turns writing numbers in each of the 2008220082 positions. Once the matrix is filled, Alex wins if the determinant is nonzero and Kiko wins if the determinant is zero. If Alex goes first, does either player have a winning strategy?
Question 16. (25 points) Consider the vector space of all (real) matrices.
(a) (12 points) Prove that the subset
is a subspace and compute its dimension.
(b) (12 points) Prove that the subset
is a subspace and compute its dimension.
(c ) (1 points) Write a haiku about linear algebra.