5. Orthongonalization and least squares [2+3+3pt]. (a) Given any two nonzero vectors x and y in R^n, construct a Householder matrix H, such that Hx is a scalar multiple of y. Is the matrix H unique? (b) Use Householder matrices to compute the QR-factorization of the matrix: (c) We believe that a real number Y is approximately determined by X with the model function Y = a exp(X)+bX² + cX + d . We are given the following table of data for the values of X and Y: Using the above data points, write down 7 equations in the four unknowns a, b, c, d. The least squares solution to this system is the best fit function. Write down the normal equations for this system, solve them in MATLAB. Plot the data points (X,Y) as points and the best fit function.

\text { (1) If } A, B \text { are bounded sets of real numbers and } A \cap B \neq \varnothing \text { , show that } \inf (A \cap B) \geq \max \{\inf A, \inf B\}

(5) The sequence {an} is defined by a_{n}=\frac{3 n-1}{n+1} \sin ^{2}\left(\frac{n \pi}{6}+\frac{1}{n}\right) Find the upper and lower limits of the sequence and all the limit points of {an :n = 1,2,.}.

(1) Use the definition of convergence of sequences to verify the following limits: \text { (a) } \lim _{n \rightarrow \infty} \frac{(-1)^{n} n}{n^{2}+1}=0 \text { (b) } \lim _{n \rightarrow \infty} n\left(\sqrt{1+\frac{1}{n}}-1\right)=\frac{1}{2} \text { . }

Consider the following ordinary differential equation (ODE): \frac{d u}{d t}=g(u) To solve this numerically, you can use the backward Euler method, for some time step At > 0 \frac{u^{n+1}-u^{n}}{\Delta t}=g\left(u^{n+1}\right) The numerical result from this process is the sequence uº, u', u², ..., which can be interpreted as an approximation to the exact solution sampled at times 0, delta t,2 delta t, . .. (a) If g(u) is a general nonlinear function and is differentiable, write down an iteration which determines un+1 from Newton's method. (b) The convergence of Newton's method depends on the choice of the initial guess. What would be a sensible choice for an initial guess?

True or False? 1. x-a is the difference between x and a. 2. If x - a>0 then x-a is the distance between x and a. 3. If x-a < 0 then x − a is the distance between x and a. 4. x=x-0 1. Reread the definitions above two or three times, then answer the following questions. (a) Applying the definition of the absolute value, | − 5| is the distance between _______and________. (b) Draw a curly brace to indicate the distance between -5 and zero on the real number line below. Label the curly brace with the expression “| — 5|". (Which side of 0 is it on? Does this make sense?) (c) What is the distance between -5 and zero? (d) Similarly draw 5 on the same number line and label it, then complete the statements below. By the definition of the absolute value |5| = ____________because the distance between _________and _________ is _____. The absolute value - 5 = _________because the distance between _____________ and _______is ___________. 2. By definition, |x| = 6 means that the distance between ________ and 0 is __________. On the number line, indicate all possible values of x that solve the equation |x| = 6. 3. We saw that |x| = 6 means the distance between x and 0 is 6. We also noted that |x| = 6 is equivalent to |x − 0| = 6. (a) Interpret the meaning of |x − 4| = 6 and write this statement as a complete sentence. (b) Mark all values that x that satisfy the sentence you wrote in (a) and explain your reasoning to your group. (c) Another way to interpret |x − 4| = 6 is to notice that |x − 4] = 6 is equivalent to [(x − 4) − 0| = 6.Then we can say that the distance between (x − 4) and 0 is _________. Here is a picture Solve for x: In which graph above did you find these same values for x? 4. Solve the absolute value equations for x and graph your solutions on a number line. \text { (a) }|5 x-2|=13 \text { (c) } 3+|4 x-1|=8 5. Absolute Value Inequalities (a) Use the definition of the absolute value to interpret the meaning of the absolute value inequality x < 6. Write your interpretation as a complete sentence. (Your sentence should start with, "The distance between...”) (b) Label the tic marks for -10, -2,0, 2, 4, 8, 6 on the number line below. Determine which of these values satisfy the sentence that you wrote in part (a). Ask yourself: Shade all values of x on the number line that satisfy the sentence you wrote in part (a) (c) Use an inequality to describe the shaded region on the number line in (b). (d) Notice that |x| < 6 is equivalent to |x − 0| < 6. Also notice that ___________ .is in the center of the shaded region above and there is a length of ___________ on either side of the center. • Indicate on the number line in (b) where we see the center and length of _____________ on each side of the center. • Draw arrows to indicate where these values are in the inequality |x − 0| < 6. 6. Interpret the inequality |x| ≥ 6 using the definition of the absolute value then shade this region on the number line. Shade all values of x on the number line that satisfy the sentence you wrote in part (a) (a) Write the region described by |x| ≥ 6 using a compound inequality: 7. Let's consider the inequality |x − 4| ≤ 5 using the definition of the absolute value. (a) First, note that |x - 4 ≤ 5 is equivalent to [(x − 4) − 0| ≤ 5.Then |x − 4| ≤ 5 means that the distance between __________________ and 0 is less than or equal to ___________. We can illustrate this on the number line: Write the compound inequality that describes the shaded region and solve for x. Graph the solution set on the real number line: (b) Look at the set of solutions for |x − 4| ≤ 5 that you found in the last exercise. What is the centre of this set? How long is the set on either side of the centre? Where do you see these in the statement of the inequality? 8. We can similarly solve the inequality |x − 4| ≥ 5. The inequality |x - 4 ≤ 5 means that the distance between _______________ and 0 is greater than or equal to_____________. We can illustrate this on the number line:

(8) Show that the following series is conditionally convergent, \sum_{n=2}^{\infty} \frac{\sin \frac{n \pi}{7}}{\ln \left(1+\frac{1}{\sqrt{2}}\right)+\cdots+\ln \left(1+\frac{1}{\sqrt{n}}\right)}

2. Suppose that a weighted graph (V, E) has transition matrix W. A new graph (V, E')is constructed from (V, E) by reversing all the edges in E. That is, if (a, b, n) E E then (b, a, n) E E' and vice versa, and the weight of the reversed edge is the same as the weight before reversing. What is the transition matrix of (V, E)? You don't need to justify your answer.

4. Sharpness of condition number estimates [4pt] Let A E R^n x n be invertible. Let b E R^n\{0},and Ax = b, Ax' = b' and denote the perturbations by Ab = b' – b and Ax = x' – x . Show that the inequality obtained in Theorem 2.11 is sharp. That is, find vectors b, Ab for which \frac{\|\Delta x\|_{2}}{\|x\|_{2}}=\kappa_{2}(A) \frac{\|\Delta b\|_{2}}{\|b\|_{2}} where k2(A) is the condition number of A under the 2-norm. (Hint: consider the eigenvectors of A^T A.)

2. Find the first two iterations of the Jacobi method and the Gauss-Seidel method for the following linear system, using (0) = 0: a. C. 4x₁ + x₂x3 = 5, -x₁ + 3x2 + x3 = -4, 2x1 + 2x2 + 5x3 = 1. 4x₁ + x₂x3 + x4 = -2, x₁ +4x2 X3 X4 = -1, x4 = 0, -x₁x₂ + 5x3 + x₁x₂ + x3 + 3x4 = 1. - b. -2x₁ + x₂ + x3 = 4, x₁-2x2x3 = x₂ + 2x3 = 0.