linear algebra, math homework help

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10 questions in total.

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c Vikram Krishnamurthy 2017 1 Assignment 1: Due May 27 Please email your solutions (as a pdf file) to the TA Sujay Bhatt at sh2376@cornell.edu Make sure your email contains your name and student number. Notation: Below x0 denotes transpose of a matrix or vector x. 1. Define what is meant by a permutation matrix. What is their purpose? Show that the product of permutation matrices is a permutation matrix. Show that the inverse of a permutation matrix is the transpose of the matrix. 2. Markowitz Portfolio Optimization: This exercise requires some elementary optimization (from your calculus course) together with basic linear algebra. Consider the following optimization problem: min x0 Cx x∈Rn 0 such that g x = r Here x is a portfolio allocation vector, C is an n × n covariance matrix, g denotes the rate of returns of individual components and the scalar r denotes the return of the entire portfolio. It is assumed that g, r, C are known. Find the optimal x. 3. Consider the equation Ax = 0 where A is an m × n matrix and 0 denotes the vector with elements 0. For the cases m < n, m = n and m > n, discuss the solutions of this equation. (Disregard the trivial solution x = 0). 4. In abstract terms, explain how the LU decomposition of a matrix works. That is: starting with a matrix A and applying successive row-wise transformations using lower triangular matrices L1 , L2 , . . . show that one ends up with upper triangular matrix U . This abstraction is at the core of solving linear systems of equations by Gaussian elimination. 5. Is a linear system of equations well posed? This is crucial - because otherwise you can get nonsensical answers. Consider solving the system Ax = b where A is an m × n matrix. Show that: (a) Either a solution x exists (b) OR there exists a vector y such that A0 y = 0, b0 y 6= 0. That is either (a) holds or (b) holds; but both (a) and (b) cannot hold. A more general statement (which arises in linear programming optimization) is: Consider solving the system Ax = b where A is an m × n matrix and x is a non-negative vector. Show that: c Vikram Krishnamurthy 2017 2 (a) Either a solution x exists (b) OR there exists a vector y such that A0 y ≥ 0, b0 y < 0. That is either (a) holds or (b) holds; but both (a) and (b) cannot hold. 6. This exercise is meant to give you some familiarity of pivoting and the so called echelon form of matrices. Compute the rank of the following matrices (justify your method)     2 0 −1 3 6 2 4 1 1 2 2 ,  2 0 2 2 0 −1 1 −1 −1 2 7. The equation of an n-dimensional plane is c0 x = a, where c is the normal to the plane and a is a scalar. Determine the point in the plane nearest the origin and find the distance of this point to the origin. 8. What is the equation of a line in n-dimensional space? 9. Show that the following matrix rotates vectors counter-clockwise (for example in computer graphics)   cos θ − sin θ 0 R =  sin θ cos θ 0 0 0 1 ...
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stephenerdberg
School: UIUC

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Question 1
We describe a permutation matrix as a square binary matrix that has exactly one entry of 1 in
each column row and each row and 0s elsewhere
a) the product of permutation matrices is a permutation matrix as shown below
0
[
1

(0 ∗ 0) + (1 ∗ 1)
1
0 1
]∗[
]=[
(1 ∗ 0) + (0 ∗ 1)
0
1 0

(0 ∗ 1) + (1 ∗ 0)
0 1
]=[
]
(1 ∗ 1) + (0 ∗ 0)
1 0

a) the inverse of a permutation matrix is the transpose of the matrix as shown here:
0
[
1

=[

1
𝑎
]∗[
0
𝑐

1
𝑏
]=[
0
𝑑

0
]
1

(0 ∗ 𝑎) + (1 ∗ 𝑏) (0 ∗ 𝑏) + (1 ∗ 𝑑)
𝑏
]=[
(1 ∗ 𝑎) + (0 ∗ 𝑐) (1 ∗ 𝑏) + (0 ∗ 𝑑)
𝑎

0
Therefore b=1, d=0, a=0, c=1 which is the transpose of [
1

1
𝑑
]=[
0
𝑐

0
]
1

1
]
0

Question 2
We re given that that g, r, and C are known.
𝑟

The first thing is to make x the subject of the formula g’ x = r to obtain x=𝑔′
Also, the condition that the portfolio weights sum to one can be expressed as

Where 1 is a 3 × 1 vector with each element equal to 1. Consider another portfolio with weights
y = (y y yc)’,

Surname: 2

The first three elements of z
 are the portfolio weights m = (  )0 for the global minimum variance portfolio with
expected return P,M = m0μ and variance 2
 = m0Σm
Question 3
A linear system with unique solution has a solution set with one element. A linear system
with no solutions has an empty set of solutions
Ax=0 is an homogeneous equation since it has a constant of zero so that it can be written as
𝑎1 𝑥1 + 𝑎2 ...

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