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$
Announcements:
Today is review.
Work through suggested exercises.
There are additional practice problems on WeBWorK and on the suggested exercises page.
Homework 11 is on Gradescope and is due today.
My next office hour is today 2:30-3:20 in MC130.
Next week: Monday, 4-5; Tuesday, 2-3; Wednesday, 1:30-2:30. All in MC130.
Review sessions next week: TBA.
Please do the course evaluations for this course!
Final Exam Preparation:
Remember that you have many resources available, including:
The exam is cumulative, but will focus on material after the midterm (Sections 3.6, 3.7, 4.1, 4.2, 4.3, 4.4, 4.6).
Sections 3.3 and 3.5 are also quite important.
Strategies for True/False questions:
- Experiment on scrap paper, to try to figure out if the statement
is true or false.
- If it is true, give an explanation. You can say things like: "By a result from
class, we know that a homogeneous system with more variables than equations
always has a nontrivial solution."
- If it is false give a specific counterexample. Try to make it small and
simple. Consider matrices like the zero matrix, the identity matrix, other
diagonal matrices, the matrix $\bmat{rr} 1 & 1 \\ 0 & 1 \emat$,
other triangular matrices, etc.
(Triangular matrices has easy eigenvalues and one easy eigenvector.)
- When looking for counterexamples, one heuristic is that a "random"
matrix is almost always invertible and is often diagonalizable.
So it's harder to come up with matrices that aren't invertible or diagonalizable.