Mathematics 244, section 1 – Linear Algebra Final Examination May 10, 2004

Directions

Do problems I - VII, and either VIII or VIII’ in the blue exam book. If you wish, you may submit solutions for both VIII and VIII’ 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 V = P_{4}(R), and let W = {p(x) ∈ V : p′(1) = 0 and p(0) = 3p(2)}. A) (10) Show that W is a vector subspace of V. B) (10) Find a subset S ⊂ W such that W = Span(S). C) (10) Is your set S linearly independent? Justify your assertion.

II. (20) Let V be a vector space. Show that a set S ⊂ V is linearly dependent if and only if there is some x ∈ S such that Span(S − {x}) = Span(S).

III. All parts of this problem refer to the linear mapping T : R^{4} → R^{3} defined by

T (x_{1}, x_{2}, x_{3}, x_{4}) = (4x_{1} − 3x_{2} + 2x_{3} − x_{4}, x_{2} + x_{3}, x_{1} − x_{4})

A) (10) Find the matrix of T with respect to the standard bases in the domain and target. B) (15) Find the matrix of T with respect to the basis

α′ = {(1, −1, 0, 0), (1, 1, 0, 0), (0, 0, 2, 3), (0, 0, −1, 2)}

in the domain and β′ = {(0, 0, 1), (0, −1, 1), (−1, 1, 0)} in the target. (You may use any applicable method.)

IV. (20) The following matrix is the augmented matrix for a system of 3 linear equations in 4 unknowns. For which value(s) of a ∈ R does the system have a solution? Find all solutions for those a.

( \begin{matrix}−1 & 2 & 0 & 1 & | & 2 \\ 3 & 1 & 3 & 0 & | & 0 \\ 1 & 12 & 6 & 5 & | & a \end{matrix} )

V. (20) Show by mathematical induction that the determinant of an n×n upper-triangular matrix A = (a_{ij}) is the product of the diagonal entries: det(A) = a_{11}a_{22} ⋯ a_{nn}.

VI. Let T : R^{3} → R^{3} be the linear mapping whose matrix with respect to the standard basis is

M = ( \begin{matrix}2 & 9 & 0 \\ 1 & 2 & −1 \\ 0 & 0 & 5 \end{matrix} )

A) (10) Find the eigenvalues of T. B) (15) For each eigenvalue λ, find a basis of E_{λ} = Ker(T − λI). C) (10) Is T diagonalizable? Explain.

VII. True - False. For each true statement give a brief proof. For each false statement give a counterexample or a reason. A) (10) If y, z are two vectors in R^{n}, then the mapping T : R^{n} → R^{n} defined by

T (x) = ⟨x, y⟩y + ⟨x, z⟩z

is linear. B) (10) No matter what the entries a, b, c ∈ R are, all the roots of the characteristic

polynomial of the matrix A = ( \begin{matrix}a & b \\ b & c \end{matrix} ) are real.

C) (10) If T : V → W is linear, dim(V) = 5, and dim(W) = 2, then for every y ∈ W the general solution of the equation T (x) = y contains three arbitrary constants.

VIII.

A) (10) Show that there exists a basis {p_{1}(x), p_{2}(x), p_{3}(x)} for V = P_{2}(R) satisfying

\begin{matrix}p_{1}(0) = 1, & p_{1}(1) = 0, & p_{1}(2) = 0 \\ p_{2}(0) = 0, & p_{2}(1) = 1, & p_{2}(2) = 0 \\ p_{3}(0) = 0, & p_{3}(1) = 0, & p_{3}(2) = 1 \end{matrix}

Be sure to say how you know your polynomials form a basis for V. B) (5) Suppose you knew that the polynomial q(x) satisfied q(0) = 8, q(1) = −3 and

q(2) = 4. What would the expansion of q(x) in terms of the basis {p_{1}(x), p_{2}(x), p_{3}(x)} be?

C) (5) Generalize what you did in part A to show that given any three distinct real numbers a, b, c, there exists a basis {p_{1}(x), p_{2}(x), p_{3}(x)} of V = P_{2}(R) satisfying

\begin{matrix}p_{1}(a) = 1, & p_{1}(b) = 0, & p_{1}(c) = 0 \\ p_{2}(a) = 0, & p_{2}(b) = 1, & p_{2}(c) = 0 \\ p_{3}(a) = 0, & p_{3}(b) = 0, & p_{3}(c) = 1 \end{matrix}

VIII’. A) (5) Show that if x is an eigenvector of a matrix A with eigenvalue λ, then x is also an

eigenvector of A^{2}. B) (5) Suppose that A is symmetric with distinct eigenvalues λ_{1}, …, λ_{k}. Let P_{1}, …, P_{k}

be the matrices of the orthogonal projections onto the eigenspaces E_{λ_{1}}, …, E_{λ_{k}} of A (with respect to the standard basis). Explain why

A^{2} = λ_{1}^{2}P_{1} + ⋯ + λ_{k}^{2}P_{k},

using the Spectral Theorem. C) (10) Verify that the formula in part B is true in the case of the 2×2 symmetric matrix

A = ( \begin{matrix}4 & 1 \\ 1 & 4 \end{matrix} ).

Have a safe, enjoyable, and productive summer!