2. Consider the following three vectors in R: V1 where ε 10-8 (a) Using floating-point arithmetic...
1. Consider the following three vectors in R Vi (1,-1,-11), v2 (3,0,-3,2), v3- (4,0,-2,2) (a) Perform the Gram-Schmidt process to find an orthonormal basis [ei,e2,e3j of the subspace spanned by {vi, V2, V3) (b) Find the QR decomposition of the following matrix A QR: 412 922 231 12 113 q13 q23 43300 14 924 934 -1 0 0 0 122 r23 Relate (rij] to the Gram-Schmidt process. (c) Can you say anything about either Qor without calculation? Show that ATA...
(1 point) Perform the Gram-Schmidt process on the following sequence of vectors. 4 -3 5 8 0 -5 X= y = ,Z = 8 -3 -2
3. Consider the following vectors, where k is some real number. H-11 Lol 1-1 a. For what values of k are the vectors linearly independent? b. For what values of k are the vectors linearly dependent? c. What is the angle (in degrees) between u and v? 4. Here are two vectors in R". Let V = the span of {"v1r2} a. Find an orthogonal basis for V (the orthogonal complement of V). b. Find a vector that is neither...
6. Consider the following algorithm, where P is an array containing random numbers. The function swap(v1,v2) will swap the values stored in the variables v1 and v2. Note that % is the modulus operation, and will return the integer remainder r of a/b, i.e., r-a%b Require: Array P with n > 0 values 1: i-1, j-n-l 2: while i<=j do for a=i to j by i do 4: 5: 6: 7: if Pla>Pat 11 and Pla]%2--0 then swap(Plal, Pla+1l) end...
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
Consider a system of three point charges as indicated in the figure below, where an x-axis and y-axis are shown as reference in the system. The coordinates of these charges are given as follows: ?1(0,0), ?2(d,0), and ?3(d,d). The charges ?1 and ?3 are positive, whereas the charge ?2 is negative. a) Express the resultant force ?⃗ 3 exerted on ?3, in terms of ?1, ?2, ?3, and d, in unit vector notation. Show all vectors on the figure, and...
Consider the following unity feedback system for Problems 2-3 R(9) —tqKAG YIS) Figure 1 Problem 2 Consider the system shown in the above figure, where G(s) = s(8+1128+1) a) Draw a Bode diagram of the open-loop transfer function G(s) when K=1. b) On your plot, indicate the crossover frequencies, PM, and GM. Is the closed-loop system stable with K=1? c) Determine the range of K for which the closed-loop systems will be stable. d) Verify your answer in (c) using...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Consider the following assembly code. 1. 1, LOAD R, #1 2, LOADS, #1 3, LOAD T, #(k-3) 4. ADD AC, R, S 5. LOAD R, S 6. LOAD S, AC 8. BRP 4, T 9. STOR AC, M where R, S, T, AC are is addressing and BRP stands for "branch if positive". sters, M is a memory location, # indicates immediate (a) Explain what this code computes (assuming that k is a natural number greater than two). (6 marks)...
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8....