M340L – Matrices Midterm Summer 2022
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While this is a take-home exam, I ask that you do the work on this exam yourself. You may ask clarifying questions on Piazza, but please don’t ask how to do the problems.
Calculators are allowed, but in order to receive credit, you must show your work. You must also justify all conclusions you make. Do not assume something is obvious. If you feel something is clear enough to not necessitate algebra, write a sentence or two explaining your reasoning. Do not do computations in your head. Instead, write them out on the exam paper.
If you are unable to simplify an answer, leave it in a form that can be put in to a calculator. Do not assume that every answer will be simple.
M340L – Matrices Midterm Summer 2022
1 (8 points) Determine if the following sets of vectors are linearly independent or not. Justify your answers.
1 2 0 (a) 43 , 21 , 65
1 2 3 (b) −1 , −3 , −2
− 3 3 0 400
5 3 1 2
(c) −2, −4, 3, 1 1 2 2 −2
M340L – Matrices Midterm Summer 2022
2 (8 points) Find and use the LU decomposition in order to solve the system Ax = b:
2 3 2 −3 −1 −6 −6 −5 9x=9 −4−9−37 4 0−312 4
Make sure to show all operations when computing the decomposition.
M340L – Matrices Midterm Summer 2022
3 (8 points) Using row reduction as done in class, find the inverse of the following matrix:
4 −3 −3 A=2 −1 0
−3 2 1 Use the matrix inverse to solve the system
Ax = 3 −3
M340L – Matrices Midterm Summer 2022
4 (8 points) Define the linear transformation T : R2 → R3 by T (x) = Ax, where
3 −2 A=0 1
What is the domain of this transformation? What is the codomain? Is this transformation one-to-one? Why or why not? Is it onto? Why or why not? What is T 32?
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