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The midterm ii exam for math 232 at sfu in spring 2012. The exam covers various topics in linear algebra, including vector multiplication, matrix operations, determinants, eigenvalues, and linear transformations. Students are required to solve problems related to computing vector products, finding determinants, and applying linear transformations to vectors. The exam consists of five questions, each worth a specific number of points, and includes instructions for the students on how to write their answers and complete the exam.
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MATH 232 D100 Spring 2012 Instructor: D. J. Katz March 14, 2012, 11:30 a.m. – 12:20 p.m.
Name: (please print) family name given name
SFU ID: @sfu.ca student number SFU-email
Signature:
Instructions:
√ (^2).
Question Maximum Score
, and v =
[2] (a) Compute u × v.
[2] (b) Compute (u × u) × v.
[2] (c) Compute u × (v × u).
[2] (d) Find the area of the parallelogram determined by u and v, i.e., where two of the sides are u and v with their initial points at the origin.
time. The two states are the ground state and the excited state. We measure the fraction of the molecules in each of the two states once per minute. Each minute, 1 / 8 of the molecules in the ground state change to the excited state, and 1 / 2 of the molecules in the excited state change to the ground state. We begin the experiment at time t = 0 minutes with 2 / 3 of the molecules in the ground state and 1 / 3 of the molecules in the excited state. [4] (a) Write a transition matrix A for this phenomenon.
[3] (b) Compute the fraction of molecules that are in the excited state one minute after the beginning of the experiment.
[4] (c) Find the long-term limit (as t → ∞) of the fraction of molecules in the excited state.
[2] (a) What is the domain of T? What is the codomain of T?
[3] (b) Find the matrix [T ], such that T (x) = [T ]x for all x ∈ R^3.
[2] (c) Find T (2, 1 , −5).
[4] (d) Find a vector x such that T (x) = (3, 1 , 2).
[3] (e) Is T an isometry (also known as an orthogonal operator)? Justify your answer.