CSE1004 ACM-1 Assignment 01 (Linear Algebra) PDF

CSE1004: ACM-1 Assignment 01 (Linear Algebra) Instructions: Grading Methods: Assignment submission is considered complete if you fol- low all instructions given below. Deadline for doubt clarification: Sep 20, 2024. – If a topic is not covered, then do self-s...

CSE1004: ACM-1 Assignment 01 (Linear Algebra) Instructions: Grading Methods: Assignment submission is considered complete if you fol- low all instructions given below. Deadline for doubt clarification: Sep 20, 2024. – If a topic is not covered, then do self-study. – No extra time will be given. Deadline for submission on Google Drive: Sep 22, 2024. – Create one PDF file of your assignment and upload it to Google Drive https://forms.gle/ChRtXTwehk8BR6Nt5. – Put all your details at the top of the first page – Name, PRN, Subject Name, Division, and Program. – Put your PRN on each page. The date and time for your viva will be scheduled after MidTerm exam. The viva schedule will be announced later. Once we have reviewed your assignment and conducted the viva, we will add our signature to your assignment. Upload a scanned copy of the signed document on Moodle before the next submission deadline. The next deadline will be announced later. Failure to follow any instructions will result in a score of 0 out of 10. Questions 1. For which values of c does the following systems (i) (ii) x + 2y + cz = 1 x+y+z =3 −x + cy − z = 0 x + (c + 1)y + 2z = 5 cx − 4y + cz = −1 2x + (c + 2)y + (c2 − 6)z = c + 11 have: (a) no solutions; 1  (b) infinitely many solutions; (c) a unique solution? Find the solutions when they exist. 2. Determine the inverse (if it exists) of each of the following matrices using elementary row operations:   1 4 1 B = 2 3 1 1 −7 −2 3. Reduce the following matrix to: (a) Echelon Form (b) Row Reduced Echelon Form using elementary row operations and determine the rank.   2 −4 3 1 0 1  −2 1 −4 2  0 1 −1 3 1 4 −7 4 −4 5 4. Find the characteristic equation, eigenvalues, and eigenvectors of the fol- lowing matrices:   −9 4 4 A =  −8 3 4 −16 8 7   2 4 −6 B= 4 2 −6  −6 −6 −15   2 −1 1 C = −1 2 −1 1 −1 2
5. Given that A is a 3 × 3 matrix, and P = v1 v2 v3 , where v1 , v2 , v3 ∈ R3. Suppose that P is invertible and the following equations hold: Av1 = 3v1 , Av2 = 2v1 + 3v2 , Av3 = v1 − 2v2 + 3v3 Find a matrix M such that AP = P M. Show that M is unique. 2  6. Test the consistency of the following systems of equations. And find the solution if it exists. System 1: 2x + 6y = −11 6x + 20y − 6z = −3 6y − 18z = −1 System 2: 4x − 2y + 6z = 8 x + y − 3z = −1 15x − 3y + 9z = 21 System 3: x+y+z =4 2x + 5y − 2z = 3 7. Let V be the first quadrant in the xy-plane; that is, let    x V = | x ≥ 0, y ≥ 0. y (a) If u and v are in V , is u + v in V ? Why? (b) Show that V is not a vector space. (Hint: Find a specific vector u ∈ V and a specific scalar c such that cu is not in V. This is enough to show that V is not a vector space). 8. Let W be the union of the first and third quadrants in the xy-plane. That is, let    x W = | xy ≥ 0. y (a) If u ∈ W and c is any scalar, is cu in W ? Why? (b) Show that W is not a vector space. (Hint: Find specific vectors u and v in W such that u + v is not in W. This is enough to show that W is not a vector space). 9. Construct a geometric figure that illustrates why a line in R2 , not passing through the origin, is not closed under vector addition. 10. Let       1 2 4 v1 =  0  , v2 = 1 , v3 = 2. −1 3 6 Let   8 w = 4. 7 Is w in the subspace spanned by {v1 , v2 , v3 }? Why? 3 11. Determine which sets in Exercises 1–8 are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers.       1 1 1 (a)  0  , 1 , 1 −2 0 1       1 3 −3 (b)  0  ,  2  , −5 −2 −4 1         2 2 1 5 (c) 0 , 2 , 1, 1 3 1 1 3 12. Let       7 4 1 4 −7 −5 −9 , v1 =    2 , v2 =    3 . v3 =   −5 5 4 It can be verified that v1 − 3v2 + 5v3 = 0. Use this information to find a basis for H = Span{v1 , v2 , v3 }. 13. Norm: Compute the norm of the vector   3 v = −4. 12 14. Inner Product: Calculate the inner product of the vectors     1 4 u = 2 and v = 5. 3 6 15. Orthogonality: (a) Determine if the vectors     1 −2 u= and v = 2 1 are orthogonal. (b) Find an orthogonal basis for the subspace spanned by the vectors     1 1 v1 = and v2 =. 1 −1 Note: An orthogonal basis is a basis in which all vectors are orthog- onal to each other. 4