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MATH 106 Fall 2023 Assignment #1 Submission Instructions It is recommended that your work be submitted in PDF format. You can handwrite and then scan your work to produce a PDF, or use software which will save files in PDF format. Image files PNG, TIFF and JPEG are also possible. Submission Deadline: Wednesday, September 20, at 11:59pm EST Read This Thoroughly • Your assignment will be submitted, graded and returned via Crowdmark. You must submit your work for each question to the correct section in Crowdmark for that question. It is possible to submit multiple pages for a question. If your question is submitted to an incorrect section in Crowdmark, it will not be marked. • This assignment consists of seven questions, and thus it will be submitted to seven sections within Crowdmark. • Be sure to start each new question on a new page. • After submitting your assignment, it is your responsibility to make sure that the pages are rotated correctly and that they are legible. • Please don’t print out the sheet of problems and try to squeeze your answers on to it. • Assignments may be submitted anytime after you receive the assignment email from Crowd- mark Mailer and before the due date/time. If you cannot locate this email from Crowdmark Mailer, it is your responsibility to contact me to help resolve the issue before the due date. What to do if I need help? • Come to Professor Anila’s office hours • Post on Piazza • Ask questions during tutorial • Form a study group (but remember everyone must submit their own work)

Marks will be deducted for solutions that omit steps or do not provide sufficient explanation. 1. [9 marks] Let ⃗u =   1 2 0   and ⃗v =   3 − 2 − 1   be vectors in R 3 . Find the following. (a) 2 ⃗u − ⃗v (b) ⃗u · ⃗v (c) ⃗u · ( ⃗v × ⃗u ) (d) A vector equation and parametric equations for the line through the points A (1 , 2 , 0) and B (3 , − 2 , − 1). 2. [8 marks] Let ⃗u = 1 3 and ⃗v = 1 1 be vectors in R 3 . (a) Find || ⃗u || and || ⃗v || . (b) Find the distance between ⃗u and ⃗v . (c) Find the cosine of the angle θ between ⃗u and ⃗v . Keep your answer in exact terms. (d) Find proj ⃗v ⃗u . 3. [8 marks] Find the scalar equation of the plane containing the points A (1 , 0 , 0), B (4 , − 1 , 7), and C ( − 2 , 0 , 3). 4. [7 marks] Answer the following questions about the given augmented matrix h A | ⃗ b i of a system of equa- tions.   1 − 2 − 1 3 0 − 2 4 5 − 5 3 3 − 6 − 6 8 2   (a) How many variables are in the system? How many equations? (b) Use Gaussian Elimination to find the RREF of the matrix. Show each step. (c) What is the rank of the augmented matrix h A | ⃗ b i ? What is the rank of the coefficient matrix A ? (d) Describe the solution to the system of equations. 5. [8 marks] Let ⃗u, ⃗ x 1 , ⃗ x 2 , . . . ⃗ x k ∈ R n . (a) What does it mean for ⃗u to be a linear combination of ⃗ x 1 , ⃗ x 2 , . . . ⃗ x k ? (b) Show that ⃗u = 2 3 is a linear combination of the vectors ⃗ x 1 = 1 1 and ⃗ x 2 = − 1 4 .