Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 8

To be submitted by March 23, 12:00

1. Let W be the subspace of 𝔽^{4} defined by the equation x_{1} + x_{2} + x_{3} + x_{4} = 0. Find the orthogonal projection of (1, 0, 0, 0) on W. Remark: I would allow here solutions that use Maple (or other software) if you include a print out of the commands you used.

2. (A). Develop a formula for the best linear approximation for a series of points

(x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n}),

where x_{i}, y_{i} ∈ ℝ and x_{1} < x_{2} < ⋯ < x_{n} without assuming that ∑ x_{i} = 0.

(B) What line do you get for the data

(0, 2), (1, 3), (2, 3)?

3. Given a monic polynomial of degree n, f(t) = t^{n} + a_{n−1}t^{n−1} + ⋯ + a_{0}, show that there exists an n × n matrix A, such that

Δ_{A}(t) = f(t).

(Suggestion: Look at past assignments.)

4. Find an explicit formula for a_{n} in the series a_{0}, a_{1}, a_{2}, … given by

0, 1, 1, 3, 5, 11, 21, 43, ....

5. Markov Processes. Imagine a particle that can be in any one of n states

S_{1}, S_{2}, …, S_{n}.

(These may be the spin states of a particle, or the room my kid is in...) Initially, we might not know exactly the state the particle is in and have at our disposal only the probability it is in a certain state, that is, we have a probability distribution

(f_{1}, f_{2}, …, f_{n}), f_{i} ≥ 0, f_{1} + f_{2} + ⋯ + f_{n} = 1.

A certain process is now taking place (it is an example of certain stochastic processes called Markov chains). At every time t = 1, 2, 3, … the particle may change its state. We know the probability it change its state from state S_{j} to state S_{i} and construct a matrix M = (m_{ij}), where m_{ij} is the probability that the particle goes from state S_{j} to state S_{i}.

(1) Show that M is a matrix of non-negative real numbers such that for every j we have ∑_{i=1}^{n} m_{ij} = 1.

(2) Prove that the probability distribution describing the state of the particle at time n is given by

M^{n} ( f_{1} ; f_{2} ; ⋮ ; f_{n} ) .

(3) Consider the case of two states S_{1}, S_{2} where M (called the transition matrix) is given by

(\begin{matrix}1/3 & 1/2 \\ 2/3 & 1/2 \end{matrix}) .

Show that for any initial probability distribution (f_{1}, f_{2}) the limit

lim_{n→∞}M^{n} (\begin{matrix}f_{1} \\ f_{2} \end{matrix})

exists, and compute it.

6. Find the characteristic polynomial, eigenvalues λ and a basis for E_{λ} for the following matrices:

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , (\begin{matrix}1 & 4 \\ 2 & 3 \end{matrix}) ,

over the fields: (i) ℝ, (ii) ℂ, (iii) ℤ/2ℤ, (iv) ℤ/5ℤ.