Homework 2

1

Given that a linear system in the unknowns \(x_{1}, x_{2}, x_{3}, x_{4}\) has general solution \(\left(x_{2}+3 x_{4}+4, x_{2}, 2-x_{4}, x_{4}\right)\) for free variables \(x_{2}, x_{4}\), find a minimal reduced row echelon for this system.

2

Use the technique of Example 2.10 in your textbook to balance the following chemical equation:

\[ \mathrm{C}_{8} \mathrm{H}_{18}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} . \]

3

Express the following functions, if linear, as matrix operators. (If not linear, explain why.)

  1. \(T\left(\left(x_{1}, x_{2}\right)\right)=\left(x_{1}+x_{2}, 2 x_{1}, -x_{1}+4 x_{2}\right)\)

  2. \(T\left(\left(x_{1}, x_{2}\right)\right)=\left(x_{1}+x_{2}, 2 x_{1} x_{2}\right)\)

  3. \(T\left(\left(x_{1}, x_{2}, x_{3}\right)\right)=\left(2 x_{3},-x_{1}\right)\)

  4. \(T\left(\left(x_{1}, x_{2}, x_{3}\right)\right)=\left(x_{2}-x_{1}, x_{3}, x_{2}+x_{3}\right)\)

4

A fixed-point of a linear operator \(T_{A}\) is a vector \(\mathbf{x}\) such that \(T_{A}(\mathbf{x})=\mathbf{x}\). Find all fixed points, if any, of the linear operators in the previous exercise.

5

A linear operator on \(\mathbb{R}^{2}\) is defined by first applying a scaling operator with scale factors of 2 in the \(x\)-direction and 4 in the \(y\)-direction, followed by a counterclockwise rotation about the origin of \(\pi / 6\) radians. Express this operator and the operator that results from reversing the order of the scaling and rotation as matrix operators.

6

Find a scaling operator \(S\) and shearing operator \(H\) such that the concatenation \(S \circ H\) maps the points \((1,0)\) to \((2,0)\) and \((0,1)\) to \((4,3)\).

7

Given transition matrices for discrete dynamical systems

  1. \(\left[\begin{array}{rrr}.1 & .3 & 0 \\ 0 & .4 & 1 \\ .9 & .3 & 0\end{array}\right] \quad\) (b) \(\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right] \quad\) (c) \(\left[\begin{array}{rrr}.5 & .3 & 0 \\ 0 & .4 & 0 \\ .5 & .3 & 1\end{array}\right] \quad\) (d) \(\left[\begin{array}{rrr}0 & 0 & 0.9 \\ 0.5 & 0 & 0 \\ 0 & 0.5 & 0.1\end{array}\right]\) and initial state vector \(\mathbf{x}^{(0)}=\frac{1}{2}(1,1,0)\), calculate the first and second state vector for each system and determine whether it is a Markov chain.

8

For each of the dynamical systems of the previous exercise, determine by calculation whether the system tends to a limiting steady-state vector. If so, what is it?

9

A population is modeled with two states, immature and mature, and the resulting structured population model transition matrix is \(\left[\begin{array}{cc}\frac{1}{2} & 1 \\ \frac{1}{2} & 0\end{array}\right]\).

  1. Explain what this matrix says about the two states.
  2. Starting with a population of \((30,100)\), does the population stabilize, increase or decrease over time? If it stabilizes, to what distribution?

10

A digraph \(G\) has vertex set \(V=\{1,2,3,4,5\}\) and edge set \(E=\) \(\{(2,1),(1,5),(2,5),(5,4),(4,2),(4,3),(3,2)\}\). Sketch a picture of the graph \(G\) and find its adjacency matrix. Use this to find the power of each vertex of the graph and determine whether this graph is dominance-directed.