Mathematics 244, section 1 – Linear Algebra Final Examination May 10, 2007

Directions

Do problems I - VII, and either VIII or IX in the blue exam book. If you wish, you may submit solutions for both VIII and IX for Extra Credit. Do not place anything you wish to have considered for credit on the exam sheet. There are 200 regular points and up to 20 Extra Credit points.

I. Let

A = ( \begin{matrix}−1 & 0 & 2 & 2 & −1 \\ 2 & 1 & −1 & 2 & −1 \\ 1 & a & −5 & −8 & a \end{matrix} )

A) (10) Are there any a ∈ R such that columns 1,2,3 of A form a linearly dependent set?

If so, find such an a. If not, say why not. B) (10) Are there any a ∈ R such that the system of equations Ax = e_{1} (the first standard

basis vector in R^{3}) is inconsistent? If so, for which a is this true? If not, say why not. C) (15) Let a = 4 and find a parametrization for Nul(A).

II. A) (5) Define: The span of a set S in a vector space V. B) (10) Given: if V is a vector space, S is a finite subset of V with Span(S) = V, and

T is any linearly independent subset of V, then |T | ≤ |S|. Use this fact to show that any two bases of V must have the same number of elements.

C) (5) The result of part B provides the justification for the definition of what concept connected to vector spaces?

III. A) (15) Solve by Cramer’s Rule:

\begin{aligned}3x_{1} + x_{3} & = 3 \\ 4x_{1} + 3x_{2} − x_{3} & = 2 \\ x_{2} + 2x_{3} & = 1 \end{aligned}

B) (5) Suppose det(A) = 3, det(B) = −2, and det(C) = 4. What is det(A^{t}B^{−1}C^{2})?

IV. All parts of this problem refer to the linear mapping T : P_{3}(R) → R^{3} defined by

T (p) = ( \begin{matrix}p(−1) \\ p(0) \\ p(1) \end{matrix} )

A) (10) Show that the image of T is a vector subspace of R^{3} directly (that is, do not just

quote a general theorem). B) (10) Find the matrix of T with respect to the basis B = {1, x, x^{2}, x^{3}} in the domain

and the standard basis in R^{3}. C) (15) State the change of coordinates formula and use it to find the matrix of T with

respect to the basis B in the domain and the basis

C = \{( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 1 \end{matrix} )\}

in R^{3}. For full credit, you must show all details of the computation of any matrix inverses you need.

V. (10) Show by mathematical induction: If A = Q^{−1}BQ for some invertible Q, then for all integers n ≥ 1, A^{n} = Q^{−1}B^{n}Q.

VI. Let A be the symmetric matrix:

A = ( \begin{matrix}a & 0 & b \\ 0 & a & 0 \\ b & 0 & a \end{matrix} )

where a, b are real numbers with b > a > 0. A) (10) Show that A has eigenvalues λ = a + b, a, a − b and determine the corresponding

eigenspaces. B) (10) Find the spectral decomposition of A.

VII. True - False. For each true statement give a brief proof (saying “this is true by definition” is not sufficient!). For each false statement give a counterexample or a reason.

A) (10) If A is an n × n matrix and AQ = QD for some diagonal matrix D and some

matrix Q, then A is diagonalizable. B) (10) If A ∈ M_{4×5}(R) and dim Col(A) = 2, then there are 4 free variables in the system

Ax = 0. C) (10) If {v_{1}, v_{2}, v_{3}} is a linearly independent subset of R^{3} and V is the matrix with

columns v_{1}, v_{2}, v_{3}, then det(V) ≠ 0. D) (10) If A is a symmetric matrix and v_{1}, v_{2}, …, v_{k} are eigenvectors of A with distinct

corresponding eigenvalues, then {v_{1}, v_{2}, …, v_{k}} is an orthogonal set.

VIII. (The Cayley-Hamilton Theorem.)

A) (10) Let A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be any 2 × 2 matrix. Show that A “satisfies its own character-

istic polynomial equation” in the sense that A^{2} − Tr(A)A + det(A)I_{2} = 0_{2×2} (here I_{2} is the 2 × 2 identity matrix and 0_{2×2} is the 2 × 2 zero matrix).

B) (10) Prove that an n × n diagonalizable matrix A satisfies its own characteristic polynomial in the same sense as in part A: If

det(A − tI) = (−1)^{n}t^{n} + c_{n−1}t^{n−1} + ⋯ + c_{1}t + c_{0},

then

(−1)^{n}A^{n} + c_{n−1}A^{n−1} + ⋯ + c_{1}A + c_{0}I_{n} = 0_{n×n}.

Caution: It does not make sense to “substitute t = A” in det(A − tI)” because t is a scalar, not a matrix. The result of problem V above may be useful. (Comment: The result is also true without the extra hypothesis of diagonalizability but it is more difficult to prove then.)

IX. (Eigenvalues in calculus!) Let V = C^{∞}(R) be the vector space of all functions f : R → R that have derivatives of all orders at all x ∈ R.

A) (10) Let D : V → V be the linear mapping defined by D(f) = f′. Show that for all

λ ∈ R, the eigenspace Ker(D − λI) has dimension 1, and find a basis. B) (5) Let S : V → V be the linear mapping defined by S(f) = f′′. Let λ = −k^{2}

for an arbitrary k > 0. Find two linearly independent functions in the eigenspace Ker(S − λI).

C) (5) Let S : V → V be the linear mapping defined by

S(f) = f(1) + f′(1)(x − 1) + \frac{f′′(1)}{2}(x − 1)^{2}.

Show that Span{1, x, x^{2}} ⊆ Ker(S − I).

Have a safe, enjoyable, and productive summer!