Problem Set 0: Math Review
Question 1: Matrix and Vector Multiplication
Calculate the following matrices and matrix-vector multiplication. (No need to hand this question in. This is just for your own practice.)
Question 1.1
\(\begin{bmatrix}3 & 1\\ 2 & 4 \end{bmatrix} \begin{bmatrix}2 & 1 & 0 \\ 1 & 5 & 2 \end{bmatrix}\)
Question 1.2
\(\begin{bmatrix}0 & 0 & 1 \\ 3 & 1 & 1 \\ -2 & 3 & 0 \end{bmatrix}\begin{bmatrix}2 & 0 & 5 \\ 1 & 3 & 1 \\ 4 & 1 & 2 \end{bmatrix}\)
Question 1.3
\(\begin{bmatrix}5 & 3 & 2 \end{bmatrix}\begin{bmatrix}2 & 7 & 1 \\ 0 & 4 & 5 \\ 2 & 3 & 4 \end{bmatrix}\)
Question 1.4
\(\begin{bmatrix}2 & 3 \\ 1 & -2 \end{bmatrix}\begin{bmatrix}2 \\ -1 \end{bmatrix}\)
Question 2: Solving Matrix Equations
Being formal and explicit in the rules of matrix algebra (e.g. when things are commutative, distributive, when you require invertibility, etc.) solve the following equations for \(x \in {\mathbb{R}}^N\), with vector \(b \in {\mathbb{R}}^N\), matrices \(A,D,Q,R\) all \({\mathbb{R}}^{N\times N}\), and scalar \(m \in {\mathbb{R}}\).
Question 2.1
\(A x + \left(x^\top D\right)^\top = b\)
Question 2.2
\(Q^{-1} \left( A\, x\, m + m b\right) = R x\)
Question 3: Linear Systems and Matrix Form
Transform the following linear equations into a linear system with matrices/vectors.
Question 3.1
\(\left\{\begin{matrix}2x + 3y = 2 \\ x - 2y = -1\end{matrix}\right.\) where \(\left\{{x,y}\right\}\) are variables
Question 3.2
\(\left\{\begin{matrix}2x - 3y = 1 \\ 3x + my = -2\end{matrix}\right.\) where \(\left\{{x,y}\right\}\) are variables
Question 3.3
\(\left\{\begin{matrix}2a + b = 1 \\ 3b + 4c = 2 \\ -2a + c = 0\end{matrix}\right.\) where \(\left\{{a,b,c}\right\}\) are variables
Question 3.4
\(\left\{\begin{matrix}a + b = -3 \\ c - 2 - 4 b = 0\end{matrix}\right.\) where \(\left\{{a,b,c}\right\}\) are variables (this will not be of full rank)
Question 4: Linear Transformations
Question 4.1
Find a linear transformation \(G \in {\mathbb{R}}^2\) such that \(G \cdot \begin{bmatrix}a & b\end{bmatrix}^\top\) always returns the second element, \(b\).
Question 4.2
Find \(H \in {\mathbb{R}}^{1 \times 2}\) such that \(H \begin{bmatrix}x & y\end{bmatrix}^\top\) returns the sum \(x + y\).
Question 4.3
Find a \(2 \times 2\) matrix \(M\) such that \(M \begin{bmatrix}p & q\end{bmatrix}^\top = \begin{bmatrix}q & p\end{bmatrix}^\top\) (i.e., swaps the two elements).
Question 4.4
Let \(T\) be a linear transformation from \({\mathbb{R}}^2\) to \({\mathbb{R}}^2\) that doubles the first coordinate and leaves the second unchanged. Write a matrix representation of \(T\).
Question 5: Orthogonality
Question 5.1
Find a vector \(x \in {\mathbb{R}}^2\) such that \(x \cdot \begin{bmatrix}1 & 2\end{bmatrix}^\top = 0\).
Question 5.2
Find a vector \(x \in {\mathbb{R}}^2\) such that \(x \cdot \begin{bmatrix}0 & 1\end{bmatrix}^\top = 1\).
Question 5.3
Given vectors \(x^1\) and \(x^2\) which may have different norms (i.e., lengths), how can you use the norm and inner product to test if they are orthogonal? Collinear?
Question 6: Undetermined Coefficients
Use undetermined coefficients to solve the following functional and difference equations.
Remember the notation \(f'(z) \equiv \frac{d f(z)}{d z}\). You don’t need to know anything about differential equations to do this problem.
Question 6.1
Take a simple linear ODE: \(f'(z) = f(z)\). Guess that \(f(z) = C_1 e^z + C_2\) and use undetermined coefficients to solve for \(C_1\) and \(C_2\).
Question 6.2
Take the functional equation \([f(z)]^2 = z^2 + 2z + 1\). Guess that the solution is of the form \(f(z) = C_1 z + C_2\). Use undetermined coefficients to find \(C_1\) and \(C_2\).
Question 6.3
Take the difference equation \(z_{t+1} = g z_t\). Guess \(z_t = C_1 C_2^t + C_3\). Show that \(C_1\) is indeterminate and find \(C_2\) and \(C_3\). What if we add subject to \(z_0 = A\)? Show how this pins down \(C_1\).
Question 7: Probability and Expectations
Let \(X\) and \(Y\) be random variables such that \(X \in \left\{{0,1}\right\}\) and \(Y \in \left\{{1,2}\right\}\). These are correlated such that
\[ \begin{aligned} {\mathbb{P}_{}\left( {X = 0 \text{ and } Y = 1} \right)} &= 0.1\\ {\mathbb{P}_{}\left( {X = 0 \text{ and } Y = 2} \right)} &= 0.3\\ {\mathbb{P}_{}\left( {X = 1 \text{ and } Y = 1} \right)} &= 0.4\\ {\mathbb{P}_{}\left( {X = 1 \text{ and } Y = 2} \right)} &= 0.2 \end{aligned} \]
Make sure to show the correct setup with numbers in the equations, but I don’t need to see intermediate steps in the calculation after that.
Question 7.1
Calculate the conditional probabilities \({\mathbb{P}\left( {X=0}\left| {Y = 1} \right. \right)}\) and \({\mathbb{P}\left( {X=1}\left| {Y=1} \right. \right)}\).
Question 7.2
Calculate the unconditional expectations \({\mathbb{E}_{{}}\left[ {X} \right]}\), \({\mathbb{E}_{{}}\left[ {Y} \right]}\), and \({\mathbb{E}_{{}}\left[ {X Y} \right]}\).
Question 7.3
Calculate the conditional expectations \(\mathbb{E}_{}\left[{X} \; \middle| \; {Y = 1} \right]\), \(\mathbb{E}_{}\left[{X} \; \middle| \; {Y = 2} \right]\), and \(\mathbb{E}_{}\left[{X Y} \; \middle| \; {Y = 1} \right]\).
Question 7.4
Calculate \(\mathbb{E}_{}\left[{X} \; \middle| \; {Y = 1 \text{ or } Y = 2} \right]\).
Question 8: Statistical Independence
Consider a worker who may be employed or unemployed (\(E\) or \(U\)), and an economy that may be good or bad (\(G\) or \(B\)). Let \(X\) be the random variable of the worker’s employment status and \(Y\) be the random variable of the aggregate economy. Now assume we know the following probabilities:
- \({\mathbb{P}_{}\left( {X = E \text{ and } Y = G} \right)} = 0.5 + \gamma\)
- \({\mathbb{P}_{}\left( {X = U \text{ and } Y = G} \right)} = 0.1\)
- \({\mathbb{P}_{}\left( {X = E \text{ and } Y = B} \right)} = 0.3 - \gamma\)
- \({\mathbb{P}_{}\left( {X = U \text{ and } Y = B} \right)} = 0.1\)
for some parameter \(|\gamma| < 0.3\).
Question 8.1
Find conditions on \(\gamma\) for statistical independence of the individual’s unemployment and the economy’s state, and interpret.
Question 9: Constrained Optimization
Solve the following optimization problems. Please be explicit in your transformation to our canonical form of constrained optimization, and be formal with Lagrange multipliers, first order necessary conditions, inequalities, etc.
Question 9.1
\[ \max_{x} \left\{ -x^2 + 2x +3 \right\} \quad \,\text{s.t.}\,x \geq 0 \]
Question 9.2
\[ \min_{x} \left\{2x + 3 \right\} \quad \,\text{s.t.}\,x \leq 1 \]