Linear Final Exam

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.

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Question 1.  (25 points)  Suppose A is a 6 \times 6 matrix with characteristic polynomial

p_A(\lambda) = \lambda^6 - 2\lambda^5 + \lambda^4

(a) (5 points) Find all of A‘s eigenvalues and all of the associated algebraic multiplicities.

(b) (5 points) Explain how it is you know that \det A = 0.

(c ) (5 points) Compute the trace of A.

(d) (5 points) Suppose A is symmetric.  What can you conclude about the geometric multiplicities of the eigenvalues?  Write down a diagonal matrix that represents A.

(e) (5 points)  Is it possible that any of the geometric multiplicities for A‘s eigenvalues equals 5?  Is it possible that any of them equal 3?  Explain your answers.


Question 2.  (20 points)  Let h \in \mathcal{L}(V, W) where V and W are real vector spaces with dimensions \dim V = n = \dim W.

Write down as many statements as you can that are logically equivalent to the statement

h \text{ is a vector space isomorphism }

(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

\displaystyle S = \left \{ 1 + 2x + 3x^2 + 4x^3, x^3 - 5x, 5x + 3, x+x^2 \right\}

form a basis for the vector space \mathcal{P}_3?  Prove your answer.


Question 4. (25 points)  Let A be a n \times n matrix.  A generalized eigenvector of rank k with eigenvalue \lambda for the matrix A is defined to be a non-zero vector \vec{v} \in \mathbb{R}^n that is in the nullspace of (A-\lambda I)^k (but is not in the nullspace of (A-\lambda I)^{k-1}).  That is

\vec{v} \neq \vec{0}, \, (A-\lambda I)^k\vec{v} = \vec{0}

and (A-\lambda K)^{k-1}\vec{v} \neq \vec{0}

(a) (3 points)  Prove that if \vec{v} is a generalized eigenvector of rank k (with eigenvalue \lambda) for the matrix A, then \vec{w} = (A-\lambda I)^{k-1}\vec{v} is an eigenvector for A with eigenvalue \lambda.

(b) (10 points) Consider the matrix

A = \left[ \begin{array}{ccc} -1 & 4 & 0 \\ -1 & 3 & 0 \\ 0 & 0 & 2 \end{array}\right]

whose characteristic polynomial is p_A(\lambda) = (\lambda-1)^2(\lambda-2).  Hence \lambda_1 = 1 is an eigenvalue with algebraic multiplicity m_1 = 2, but one can check that the geometric multiplicity is \mu_1 = 1.  It also follows that \lambda_2 = 2 is an eigenvalue with algebraic and geometric multiplicities equal to 1.

Verify that \vec{v}_1 = (2, 1,0)^T is an eigenvector with eigenvalue \lambda_1 = 1 for A and that \vec{v}_3 = (0, 0, 1)^T is an eigenvector with eigenvalue \lambda_2 =2.

Also verify that \vec{v}_2 = (1, 1,0)^T is a generalized eigenvector with eigenvalue \lambda_1 = 1 for A.

Finally, represent the matrix A using the basis \mathcal{B} = \{\vec{v}_1, \vec{v}_2, \vec{v}_3 \}

(c ) (12 points)  Suppose A is a 4 \times 4 matrix and that the basis \mathcal{B} = \{\vec{v}_1, \vec{v}_2, \vec{w}_1, \vec{w}_2\} has the following properties:

\displaystyle (A-2I)\vec{v}_1 = \vec{0} \text{ and } (A-5I)\vec{w}_1 = \vec{0}

\displaystyle (A-2I)\vec{v}_2 = \vec{v}_1 \text{ and } (A-5I)\vec{w}_2 = \vec{w}_1

Explain why these properties imply that \vec{v}_2 and \vec{w}_2 are generalized eigenvectors (both of rank 2) with eigenvalues 2 and 5, respectively.

Lastly, represent the matrix A using the basis \mathcal{B}


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Question 5.  (25 points)  Consider the linear system of differential equations

\left( \begin{array}{c} x' \\ y' \end{array}\right) = A\,\left(\begin{array}{c} x \\ y \end{array}\right) = \left(\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right)\left(\begin{array}{c} x \\ y \end{array}\right)

(a) (5 points)  Describe the behavior of solutions in the phase plane assuming A^T = A, \det A \neq 0 and \text{tr} A = 0.

(b) (5 points) Describe the behavior of  solutions assuming A^T = A, \det A > 0 and \text{tr} A > 0.

(c ) (15 points) Describe the behavior of  solutions assuming A^T = -A and \text{tr}A = 2a_{11}.


Question 6.  (25 points)  For this problem V is a vector space with an inner product \langle \, , \, \rangle.

(a) (3 points)  Using only the definitions of inner product and vector space, prove that \vec{v} \in V is orthogonal to every vector in V \iff \vec{v} = \vec{0}.

(b) (7 points)  The orthogonal complement of a subspace U \subseteq V is defined to be

U^{\perp} = \{\vec{v} \in V : \langle \vec{v}, \vec{u} \rangle = 0 \, \, \forall \, \vec{u} \in U \}

Prove that the orthogonal complement of a subspace is itself a subspace of V.

(c ) (15 points) Suppose A \in \mathcal{L}(\mathbb{R}^n, \mathbb{R}^m) — where each \mathbb{R}^n and \mathbb{R}^m are equipped with their standard inner products.  Prove that

\mathcal{N}(A^T)^{\perp} = \mathcal{R}(A)


Question 7. (25 points)

(a) (10 points)   Define a relation \sim on the set of all n \times n matrices by

A \sim B \text{ means } \, \exists P \in \mathcal{M}_{n\times n}, \,\, P^{-1}AP = B

Prove that \sim is an equivalence relation.

(b) (8 points)  Prove that if A \sim B, then for every natural number k it follows that A^k \sim B^k.

(c ) (7 points) Suppose A is a 3 \times 3 matrix with eigenvalues, associated algebraic multiplicities and associated geometric multiplicities given by

\lambda_1 = 1, m_1 = 2, \mu_1 = 1

\lambda_2 = 4, m_2 = 1, \mu_2 = 1

Is there a diagonal matrix D that satisfies A \sim D?  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 \mathcal{M}_{n\times n}.  For this problem, you do not have to re-prove this fact, and we will use the notation h(M) = M^T for this linear transformation.

(a) (10 points)  Compute the matrix representation

\displaystyle A = \text{Rep}_{\mathcal{E}, \mathcal{E}}\,(h)

where \mathcal{E} is the standard basis for the space \mathcal{M}_{2\times 2}.

(b) (15 points) Find a basis for \mathcal{M}_{2\times 2} that diagonalizes A or explain why no such basis exists.

(c ) (Bonus) Find a basis for \mathcal{M}_{n \times n} that diagonalizes h \in \mathcal{L}(\mathcal{M}_{n\times n}, \mathcal{M}_{n\times n}) or explain why no such basis exists.


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Question 9 (20 points)

(a) (6 points) Write down a 2\times 2 matrix that cannot be diagonalized, or explain why no such matrix exists.

(b) (8 points) Write down a 2\times 2 nilpotent matrix that has two distinct eigenvalues, or explain why no such matrix exists.

(c ) (6 points) Suppose A is a 2\times 2 matrix with two distinct, real eigenvalues and whose trace \text{tr}A = -2.  Is it possible that \det A > 1?  Explain your answer.


Question 10 (25 points) Suppose A is a n \times n matrix that satisfies A^2 = A (in other words A is idempotent.)

(a) (7 points) Prove that each eigenvalue of A equals 0 or 1.

(b) (18 points) Prove that A is diagonalizable.


Question 11 (25 points) Consider the linear transformation h(f(x)) = x\,f'(x) from \mathcal{P} to \mathcal{P} (the space of all polynomials)

(a) (10 points) Compute the nullity of h by finding a basis for \mathcal{N}(h).

(b) (5 points) Find a polynomial that fails to be in the range space \mathcal{R}(h)

(c ) (10 points) Find all eigenvectors and eigenvalues of h.


Question 12 (25 points) Suppose A is a square matrix with characteristic polynomial

p_A(\lambda) = (\lambda-1)^2(\lambda-2)^3(\lambda-3)^5(\lambda-4)

(a) (2 points) Explain why A has eigenvalues \lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3 and \lambda_4 = 4.

(b) (3 points) What is the size of matrix A?

(c ) (5 points) Suppose A is symmetric.  Compute the geometric multiplicities of A‘s eigenvalues, and explain your answer.

(d) (5 points) Compute the determinant of A.

(e) (5 points) Write down the Jordan-Canonical form of A assuming \mu_1 = 2, \mu_2 = 2, and \mu_3 = 3.

(f) (5 points) Suppose there exist two non-zero vectors \vec{v} and \vec{w} that satisfy A\vec{v} = 4\vec{v} and A\vec{w} = 4\vec{w}.  Explain why \vec{v} and \vec{w} are multiples of one another.


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Question 13 (25 points) For parts (a) and (b) of this problem let V be a vector space with inner product \langle\, , \, \rangle and let \mathcal{B} = \{\vec{v}_1, \vec{v}_2, \cdots, \vec{v}_n\} be a basis for V.

(a) (10 points) Consider the set of vectors \{\vec{u}_1, \vec{u}_2, \cdots, \vec{u}_n \} that are defined by the (recursive) formulas

\vec{u}_1 = \vec{v}_1

\displaystyle \vec{u}_2 = \vec{v}_2 - \frac{\langle \vec{v}_2, \vec{u}_1\rangle}{\langle \vec{u}_1, \vec{u}_1\rangle}\vec{u}_1

\displaystyle \vec{u}_3 = \vec{v}_3 - \frac{\langle \vec{v}_3, \vec{u}_1\rangle}{\langle \vec{u}_1, \vec{u}_1\rangle}\vec{u}_1 - \frac{\langle \vec{v}_3, \vec{u}_2\rangle}{\langle\vec{u}_2, \vec{u}_2\rangle}\vec{u}_2

\displaystyle \vec{u}_4 = \vec{v}_4 - \frac{\langle \vec{v}_4, \vec{u}_1\rangle}{\langle \vec{u}_1, \vec{u}_1\rangle}\vec{u}_1 - \frac{\langle \vec{v}_4, \vec{u}_2\rangle}{\langle\vec{u}_2, \vec{u}_2\rangle}\vec{u}_2- \frac{\langle \vec{v}_4, \vec{u}_3\rangle}{\langle\vec{u}_3, \vec{u}_3\rangle}\vec{u}_3

\displaystyle \vdots

\displaystyle \vec{u}_k = \vec{v}_k - \sum_{j=1}^{k-1} \, \frac{\langle \vec{v}_k, \vec{u}_j \rangle}{\langle \vec{u}_j, \vec{u}_j \rangle}\,\vec{u}_j

Prove that the set \{\vec{u}_1, \vec{u}_2, \cdots, \vec{u}_n\} is an orthogonal set of vectors.  (Suggestion: you may first want to check that \vec{u}_1 and \vec{u}_2 are orthogonal, and then that \vec{u}_3 is orthogonal to both \vec{u}_1 and \vec{u}_2.)

(b) (5 points)  Explain why the orthogonal set \{\vec{u}_1, \cdots, \vec{u}_n\} from part (a) is still a basis for V.  Lastly, explain why the new set of vectors, \{\vec{w}_1, \dots, \vec{w}_n\} where

\displaystyle \vec{w}_i = \frac{\vec{u}_i}{\|\vec{u}_i\|}

is an orthonormal basis for V.

(c ) (10 points)  Apply the algorithm / recursive formula(s) from parts (a) and (b) to replace the basis \mathcal{B} = \{1, x, x^2\} for \mathcal{P}_2 with an orthonormal basis.  For this part we have equipped \mathcal{P}_2 with the inner product

\displaystyle \langle f(x), g(x) \rangle = \int_0^1 \! f(x)g(x) \, dx.


Question 14. (25 points)  This question is about one of my other favorite 2\times 2 matrices.  Here it is:

A = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]

(a) (2 points) Write a one sentence explanation justifying the claim that A is diagonalizable.

(b) (3 points) Here’s why A is one of my favorite matrices: you can use it to build a cool sequence of numbers, F_0, F_1, F_2, F_3, \cdots .

How is this done?  You repeatedly multiply A against (1, 0)^T.  In particular, we’ll  get this sequence going by declaring the first two numbers: F_0 = 0 and F_1 = 1.  To get the next number in this list, F_2, you use the given matrix according to the following rule:

\left(\begin{array}{c} F_2 \\ F_1\end{array}\right) = A\left(\begin{array}{c} 1 \\ 0 \end{array}\right)

This rule can be repeated to generate all the numbers in this list:

\left(\begin{array}{c} F_3 \\ F_2 \end{array}\right) = A^2\left(\begin{array}{c} 1 \\ 0 \end{array}\right)

\left(\begin{array}{c} F_4 \\ F_3 \end{array}\right) = A^3\left(\begin{array}{c} 1 \\ 0 \end{array}\right)

\left(\begin{array}{c} F_{n+1} \\ F_n \end{array}\right) = A^n\left(\begin{array}{c} 1 \\ 0 \end{array}\right)

Now for the actual question: what crazy famous sequence of numbers does this process produce?

(c ) (10 points) Find a change-of-basis matrix P so that

P^{-1}AP = \left[ \begin{array}{cc} \lambda_1 & 0 \\ 0 & \lambda_2 \end{array}\right)

and be sure to find the eigenvalues \lambda_1 and \lambda_2, too.

(c ) (10 points) Use your work in part (c ) to find a formula for the n-th number in this (famous) sequence, a formula that only uses the eigenvalues \lambda_1 and \lambda_2.


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 \mathcal{M}_{n\times n} of all (real) n \times n matrices.

(a) (12 points) Prove that the subset

V_1 = \left\{ A \in \mathcal{M}_{n\times n} : A^T = A \right\}

is a subspace and compute its dimension.

(b) (12 points) Prove that the subset

V_2 = \left\{A \in \mathcal{M}_{n\times n} : A^T = -A \right\}

is a subspace and compute its dimension.

(c ) (1 points) Write a haiku about linear algebra.

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