Download Information and Diagnostic Test for Linear Algebra, July 2016

Information and Diagnostic Test for Linear Algebra, July 2016
Richard Hoshino
Quest University Canada
Dear July Block Applicant,
I am delighted to offer a course in July in Linear Algebra, and I hope to have the opportunity to
work with you this summer.
As you might imagine, math courses at university are significantly harder than math courses in
high school, both in terms of the content and the level of maturity required to communicate
solutions effectively and rigorously. In my own teaching practice, I focus heavily on the art of
mathematical communication, both orally and through writing, as I believe this is the most
important skill that is gained through a mathematics education.
To succeed in this Linear Algebra course, students must demonstrate a sufficient level of
mathematical ability through written solutions. As a result, I ask you to complete the following
“diagnostic test” that will enable me to verify whether you are ready for Linear Algebra at Quest.
Here are the rules:
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There is no time limit.
You may consult your class notes and class textbooks.
You may not use the Internet.
You must work individually, i.e., no consulting with your teachers or classmates.
For each problem, I'd like you to write out a full solution, i.e., a clear step-by-step description of
what you did to solve the problem. I care more about the process of what you did than the final
answer itself.
Make sure you type or neatly handwrite your solutions on standard 8.5 × 11 paper, with your
entire submission saved as a single file and your name on the top of each page.
Don't be discouraged if you cannot solve all the problems. Do the best that you can on each
problem and hand in what you have. I will grade your diagnostic, and you will be eligible for
admission to Linear Algebra as long as your score is above a certain threshold.
Sincerely,
Richard Hoshino
Tutor (Professor) of Mathematics
Quest University Canada
Diagnostic Test for Linear Algebra at Quest
Question #1
A straight line passes through the points (-2,-2) and (4,1).
(a) Find the equation of this line. (It will be of the form y=mx+b for some values m and b.)
(b) This line passes through the point (6,t). Determine the value of t.
(c) Determine the equation of the line that passes through the point (2,0) that is perpendicular to the
above line.
Question #2
(a) Solve this system of two equations and two unknowns, i.e., find the unique pair (x,y) that satisfies
both equations:
2𝑥 − 𝑦 + 1 = 0
{
3𝑥 − 2𝑦 + 3 = 0
(b) Create a grid, mark the x-axis and the y-axis, and sketch a graph of the lines y=2x+1 and y=3x/2
+ 3/2. Determine the point (x,y) where these two lines intersect.
(c) Explain why your answer to part (a) matches your answer to part (b).
Question #3
The general form of a quadratic function is f(x)=ax2+bx+c, where a≠0. The graph y=f(x) is a parabola.
(a) Find a quadratic function for which f(1)=6, f(2)=11, and f(3)=18. Show your work.
(b) Create a grid, mark the x-axis and the y-axis, and label the points (1,6), (2,11), and (3,18). Draw
the parabola y = f(x), where f(x) is your quadratic function from part (a).
Question #4
(a) Carefully explain why this system of two equations and two unknowns has no solution (x,y):
2𝑥 − 𝑦 + 1 = 0
{
6𝑥 − 3𝑦 + 4 = 0
(b) BONUS: Solve this system of three equations and three unknowns:
2𝑥 + 3𝑦 + 4𝑧 = 10
{3𝑥 + 3𝑦 + 3𝑧 = 12
5𝑥 + 4𝑦 + 3𝑧 = 18
Hint: there are infinitely many solutions. Your final answer will be a set of solutions,
where each triplet (x,y,z) in your set satisfies all three equations.