Complete these 10 questions on paper and make sure you send clear photos of your work and solutions.
Math 1104C, 1104J: Test # 1, Due Feb. 9 11:59, 2022 Directives: • This document has 3 pages (including this page). • This assignment is due on March 16th at 11:59 p.m. on Brightspace. • The assignment has 10 questions for a total of 65 points. • You must show your work when appropriate. We look at lot more at your work than your final answer and we need to see your work when asked. IMPORTANT: Submit a scan or pictures of your work. You can write all your answers on a piece of paper of your choice or an electronic tablet or use latex. Instructions to submit your assignment online • You must send a scan or a picture of your work. It must clear enough so that we can read what you wrote. We are only marking what we receive so make sure to send complete files. • only .pdf and .jpg files are accepted. • Make sure to submit your work at the appropriate link of the exam on Brightspace. • Only the last version of a file or document that you submit is marked. • Please submit your work as only one file or folder • No late work will be accepted. You can contact me by email at
[email protected]. 1. (4 points) Compute the determinant of the matrix B = 2 −3 7−4 5 −14 8 −9 21 , 2. Compute the determinants of the following matrices. (a) (4 points) ∣∣∣∣∣∣∣∣∣∣ 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 ∣∣∣∣∣∣∣∣∣∣ 1 mailto:
[email protected] (b) (4 points) ∣∣∣∣∣∣∣∣∣∣ 6 2 0 0 0 −3 12 1 4 5 1 0 0 0 0 12 4 0 3 4 11 5 0 1 2 ∣∣∣∣∣∣∣∣∣∣ 3. (6 points) Let |A| = ∣∣∣∣∣∣ a b c d e f g h i ∣∣∣∣∣∣ = 8. Find the determinant of the following matrix. Explain your answer. ∣∣∣∣∣∣ g h i a + 2d b + 2e c + 2f 3d 3e 3f ∣∣∣∣∣∣ 4. (5 points) Consider the vectors ~v1 = −21 1 , ~v2 = 1−2 1 , ~v3 = 11 −2 . Are they linearly independent? If not either: (a) Find a linear dependence relation among them. or: (b) Express one of the vectors as a linear combination of the others. 5. (12 points) Let ~v1 = 12 3 , ~v2 = −12 1 , ~v3 = 1−1 1 . (a) Show that the set B= {~v1, ~v2, ~v3} is a basis for R3. (b) With B as in (a), find the coordinates for x = 01 2 in the basis B . That is, find [x]B. (c) If [x]B = 1−1 1 then what is x in the standard basis? 6. (a) (4 points) Find a basis for the subspace of R4 spanned by ~u = 1 −5 3 4 , ~v = 4 −7 2 5 , ~w = 7 4 −9 −5 . (b) (1 point) Using part (a), determine whether the vectors ~u, ~v, ~w are linearly dependent. 7. (8 points) The following matrix: Page 2 A = 1 −2 4 −1 0 5 0 −2 4 −7 1 2 −8 0 3 −6 12 −3 1 15 0 2 −4 9 −3 3 12 0 has reduced row echelon form: 1 −2 0 3 0 −3 0 0 0 1 −1 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 Find bases for the row space of A, the column space of A and the null space of A. 8. (9 points) Consider B = 3 0 40 2 7 0 0 1 (a) Find all eigenvalues of B. (b) Find a corresponding eigenvector for each eigenvalue in part (a). 9. (4 points) Determine which of the following vectors ~v = 1 1 1 −1 , ~w = 1 −1 −1 2 is an eigenvector of the matrix w = 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 and find the corresponding eigenvalue. 10. (4 points) Let A = ( 5 −6 3 −4 ) . Find the eigenvalues and eigenvectors of A. Page 3