Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81Determine whether these vectors are linearly independent or not:
1−2−1,
2−31,
5−81MAE501 Homework #2 due Sunday, September 18, 2022 at 11:59pm through Gradescope Problem 1. Encryption (12 points) Please, refer to Case Study 1 Encryption handouts for this assignment. Write MATLAB functions encrypt.m and decrypt.m that perform encryption and decryption of a message of any arbitrary size using a given square encryption key matrix of any size (encryption matrix must have integer entries, and so is its inverse). Make sure to pad the message with any characters (spaces, x, etc.) to bring it to a length necessary for encryption. Remove the added characters to recover original message after decryption. Test your code using the phrase ‘Have you seen Sparky today? Yes, at the Sun Devil Stadium.’ Execute the code with the following keys: 1. m = [ 1 1 2 1 ] 2. m = 1 2 3−1 −1 −3 9 16 28 3. m = 1 −1 3 −2 2 −1 0 −1 7 −5 10 −6 1 0 −2 4 Make sure you can correctly encrypt and decrypt the phrase. Provide published .pdf output with the functions and results for all three keys. Note: Additionally, please submit your MATLAB codes as .m file into the Dropbox link to be specified. Please save both encryption and decryption functions into the same .m file (function). Include the following line as the first line in your .m file. function[sencrypt,sdecrypt]=q1 ASUID(s,m) %%%% Read comments before submitting your file %%%% Note 1: Inputs: s is the phrase to be encrypted, m is the encryption key matrix. Outputs: sencrypt - encryption of the original phrase, sdecrypt - decryption of the encrypted phrase. All inputs and outputs are case-sensitive. %%%% Note 2: DO NOT initialize ‘s’ and ’m’ inside the code (if you have it initialized, comment it out). Evaluator can select s, m of any size. %%%% Note 3: File should be saved with name as q1 ASUID.m %%%% Code starts %%%% %%%% Code ends %%%% Example for first line: if your 10-digit ASU ID is ‘1234567890’ then first line will be function[sencrypt,sdecrypt]= q1 1234567890(s,m) and file should be saved as q1 1234567890.m Problem 2 (10 points) Verify that your LU-factorization program written in Homework 1 works for rectangular matrices. Test it on the matrices a) A = 1 −1 3 5 6 7 2 5 7 0 3 4 −3 4 0 −1 −1 0 −6 0 −10 −6 −10 11 1 b) B = 2 0 2 −1 4 −2 −3 6 3 −5 6 3 0 2 7 −8 17 −11 −1 20 1 7 5 −13 Submit your published code and answers as a .pdf. Note: Additionally, please submit your MATLAB function as .m file into the Dropbox link to be specified. Please, upload your codes even if they are identical to the codes you submitted for Homework 1. Include the following line as the first line in your .m file. function[L,U]=q2 ASUID(A) %%%% Read comments before submitting your file %%%% Note 1: L and U should be the name of output 2D arrays (Case sensitive) in code. %%%% Note 2: A is the input matrix provided by evaluator, so DO NOT initialize ‘A’ inside the code (if you have it initialized, Comment it out). Evaluator can select A of any size 6×5, 5×6 etc %%%% Note 3: File should be saved with name as q2 ASUID.m %%%% Code starts %%%% %%%% Code ends %%%% Example for first line: if your 10-digit ASU ID is ‘1234567890’ then first line will be function[L,U]= q2 1234567890(A) and file should be saved as q2 1234567890.m Problem 3 (12 points) Which of the following subsets of R3 are actually subspaces? If not, explain why. a) The plane of vectors x = (χ1, χ2, χ3) ∈ R3 with first component χ1 = 0. b) All vectors with χ1 ≥ χ3, and χ2 any real number. c) The vectors x with χ1χ2 = 0. d) The solitary vector x = (0, 0, 0). e) All combinations of three given vectors x = (1, 0, 0), y = (2, 0, 1), and z = (−3, 0, 2). f) The vectors x such that χ1 + χ3 = −1. Problem 4 (16 points) Let A = 1 2 −2 1 0 2 4 −3 3 0 −3 −6 5 4 8 5 10 −9 6 0 . 1. What are the pivots, pivot variables, and free variables? 2. What is the nullspace N(A) (use the special solutions)? 3. What is the rank of A? 4. Give the complete solution to Ax = b, where b = A 1 0 2 0 1 . Problem 5 (6 points) Write out a condition on b so that the system 2 1 00 1 2 3 [ u v ] = b1b2 b3 has a solution. Problem 6 (8 points) Find a 2 by 3 system Ax = b whose complete solution is x = 12 0 + c1 13 1 . Give both A and b. Problem 7 (4 points) Determine whether these vectors are linearly independent or not: 1−2 −1 , 2−3 1 , 5−8 1 Problem 8 (12 points) Consider the matrix A = 1 0 a 2 −1 b 1 1 c −2 1 d a) Give all vectors a b c d which will make the columns of A linearly dependent. b) Give all vectors a b c d which will make the columns of A a basis for x y z t : y + t = 0 (i.e. the subspace of R4 where y + t = 0)? c) For a b c d = 1 −2 5 2 compute a basis for the four subspaces of A. 3