Linear Algebra (Math 314) Project 1 数学代写
project – 数学代写 , 这个项目是project代写的代写题目
Due Friday, July 1
Instructions: The following project is based on several sections of our text and associated applications. The main goal is to gain a deeper understanding of various concepts via a guided step-by-step solution. You must show all relevant work on all problems in order to get full credit. You must sufficiently explain your work so that one of your classmates can follow your work.You are permitted to talk with other students while preparing your answers to these questions. However, your work must ultimately be your own and in your own words. These assignments will be monitored for plagiarism. If I suspect of plagarism, I reserve the right for oral follow-up about your written solutions.
Problem 1(Interpolating Polynomial). Suppose that experimental data are represented by a set of points in the plane. Aninterpolating polynomialfor the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. Find the interpolating polynomialp(t) =a 0 +a 1 t+a 2 t^2 for the data (1,12),(2,15),(3,16) by following the following steps.
(1) Write down the corresponding linear system (with 3 variables and 3 equations) by plugging in each of the data points. (2) Write down the RREF of the corresponding augmented matrix. (3) Write down the solution (a 1 ,a 2 ,a 3 ) to the corresponding linear system. Write down the inter- polating polynomial. (4) Estimate the partial data (2. 5 , ?) and (?,0) (that is, find?).
Problem 2(Interpolating Polynomial Continued). Letnbe any positive number, and let a 1 ,…,anbe any real numbers. Consider thennmatrix
1 a 1 a^21 a^31 ... an 1 ^1 1 a 2 a^22 a^32 ... an 2 ^1 1 a 3 a^23 a^33 ... an 3 ^1 .. .
1 an a^2 n a^3 n ... ann^1
(1) Compute det(A 2 ). (2) Show that det(A 3 ) = (a 2 a 1 )(a 3 a 2 )(a 3 a 1 ). Conclude thatA 3 is invertible if and only if … (3) Make an educated guess for det(An).
Note:The matrixAnis very closely related with the matrix you had in Problem 1 (forn= 3). This is the reason why there existsexactly onepolynomial of degree at mostn1 passing throughn differentpoints. This requires just a little bit of thinking, but in a nutshell, beacuse det(An) 6 = 0, we know thatAnwill be invertible, and thus when solving for the coefficients of the polynomial there will be a unique solution.
Problem 3(Inverses for non-square matrices). LetAbe anmnmatrix, wherem < n, and assume its rank is exactlym. Show that there exists a matrixBsuch thatAB= Im, by showing the following steps:
(1) Show that, forj= 1,…,mthe matrix equationAx=ejhas a non-trivial solution. Here, as usual,ejdenotes the vector inRmthat has 1 on entryjand zero elsewhere. (2) LetbiinRnbe a non-trivial solution ofAx=ej, and consider the matrixB= [b 1 b 2 bm]. Show thatBdoes the job, that is,AB=Im. (3) Find (show all your steps) a matrixBsuch that
[ 1 1 1 2 3 4
Note:This was a guided way to prove a not-so-easy statement. Splitting the statement into two mini statements makes the job much easier. Wouldnt you agree? Part (1) could be considered a big hint into proving the entire statement. The matrixB(that always exsits in this case) is called aright inverse. Note the use of a instead of the there could be many right inverses. A similar statement holds forleft inverses.
Problem 4(Markov Chains).In this problem we will cover (and go beyond, using what weve learnt in Chapter 5) concepts from Section 4.9. First some definitions. A vector with non-negative entries that add up to 1 is calledprobability vector. A (square) matrix for which every column is a probability vector is calledstochastic matrix. AMarkov chainis a sequence of probability vectorsx 0 ,x 1 ,x 2 ,…such that
xk+1=Pxk, k= 0, 1 , 2 , 3 ,... (0.1)
andPis stochastic matrix^1.
(1) Consider the stochastic matrixP =
1 a b a 1 b
, with 0< a,b <1. Show that= 1 is an eigenvalue ofP and find its corresponding eigenvectoru. (DO NOT PICK YOUR OWN NUMBERS!) (2) Takea= 1/ 2 ,b= 1/4. Find the other eigenvalueofPand corresponding eigenvectorv.
(3) Consider the vectorx 0 =
. Finds,tsuch thatx 0 =su+tv.
(4) Show that equation (0.1) now reads as
xk=sku+tkv, k= 0, 1 , 2 , 3 ,....
Show that the sequencex 0 ,x 1 ,x 2 ,…,xk,…converges to the vector
Note: In this specific instance, we have proved Theorem 18 on page 261. Of course the general case is much more difficult!
(^1) Note here that we are implicitly saying that ifPis a stochastic matrix andxis a probability vector thenPxis also a probability vector. Youre encouraged to convince yourself!