MATH 296 PROBLEMS 6

IGOR KRIZ

Regular problems:

1. Let

A = ( \begin{matrix}1 & 2 & −1 & 2 & 0 \\ 2 & 4 & −1 & 1 & 1 \\ 3 & 6 & −2 & 3 & 1 \\ 4 & 8 & −3 & 5 & 1 \end{matrix} ) .

(a) Find a basis of Null(A).

(b) Find a basis of Col(A).

2. Review problem: Find the shortest distance from a given point (0, b) on the y-axis to the parabola y = \frac{x^{2}}{4} . [Express the distance as a function, and find its minimum using derivatives.]

3. Let V be a vector space, let S and T be two subsets of V (not necessarily subspaces).

(a) Prove that < S ∩ T >⊆< S > ∩ < T >.

(b) Find an example where < S ∩ T >≠< S > ∩ < T >

4. Consider the set

S = \{( \begin{matrix}1 \\ 2 \\ −1 \\ 1 \end{matrix} ) , ( \begin{matrix}2 \\ 3 \\ −1 \\ 1 \end{matrix} ) , ( \begin{matrix}1 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 1 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ −1 \\ −1 \end{matrix} )\}.

Does S span ℝ^{4}? Is S linearly independent in ℝ^{4}?

Challenge problems:

5. Linear recursions continued: Suppose a sequence (x_{n}) is defined as follows: x_{0},...,x_{k−1} are given, and

x_{n} = a_{k−1}x_{n−1} + ... + a_{0}x_{n−k} for n ≥ k. (1)

(a) Suppose that the polynomial

p(x) = x^{k} − a_{k−1}x^{k−1} − a_{k−2}x^{k−2} − ... − a_{0}x^{0}

has a root γ of multiplicity ≥ k (i.e. (x − γ)^{k} divides p(x)). Show that then the sequences

x_{n} = n(n − 1)...(n − i)γ^{n}

for i < k − 1 and all their linear combinations satisfy the relation (1). [Hint: use derivatives.]

(b) Using (a), solve the following problem: Suppose numbers x_{n} are given as follows: x_{0} = 1, x_{1} = 2, x_{n+2} = 2x_{n+1} − x_{n} for n ≥ 0. Find a formula for x_{n}.

(c) Consider the sequence 1, 4, 2, 1, 4, 2, 1, 4, 2, .... Thus, x_{3n} = 1, x_{3n+1} = 4 and x_{3n+2} = 2 for all natural numbers n. Find a formula for x_{n} which uses only arithmetic operations (addition, multiplication, subtraction, division, taking powers and roots). [Write the sequence in terms of a linear recursion. This does not use (a) or (b), but it uses complex numbers.]

6. Review problem: If

C_{0} + \frac{C_{1}}{2} + ... + \frac{C_{n−1}}{n} + \frac{C_{n}}{n + 1} = 0,

where C_{0}, ..., C_{n} are real constants, prove that the equation

C_{0} + C_{1}x + ... + C_{n−1}x^{n−1} + C_{n}x^{n} = 0

has at least one real solution x ∈ (0, 1).