12-240/The Final Exam

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The Results

104 students took the exam; the average grade was 70, the standard deviation was 26, and the exam itself is at Final.pdf.

Restricted to the 101 students who took the final, the average course grade was 75 and the standard deviation was 19.


Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

Content and Style

It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:

  • Fields and vector spaces.
  • Spans, independence, replacement and bases.
  • Linear transformation, rank, nullity, matrices.
  • Row and column reduction and elementary matrices, systems of linear equations.
  • Determinants.
  • A bit on diagonalization.
  • Several other "smaller" topics.

As for the style -

  • You can expect to be asked to reproduce some proofs that were given in class.
  • You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
  • You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
  • You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See 06-240/The_Final_Exam and 09-240/The Final Exam.

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure often comes out to be .

Remember. Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "", does it mean "choose ", or "we've just proven that ", or "assume by contradiction that ", or "for every " or "there exists "? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

Office Hours

Brandon and Peter and I will hold pre-exam office hours as follows:

  • Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
  • Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
  • Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
  • Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
  • Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
  • Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
  • Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.
Dror's notes above / Students' notes below

Cheat Sheets made by Students

I don't know where to post this, but if you are doing some last minute study questions like me and you want to check the even answers (as the odd can be checked in the back of the book), here's a good link: http://www.scribd.com/doc/64217240/Linear-Algebra-Friedberg-4th-Ed-Solutions-Manual