Linear Algebra Assignment
IMPORTANT NOTES ON COLLABORATION
Let me quote from MAT137 problem sets on this issue.
Solving a mathematical problem set has two parts:
1. The discovery phase. This is the time you spent trying to figure out how to solve the problems, and it often takes most of the time. You are welcome and encouraged to collaborate with other students in this phase. Collaboration is a healthy practice, and this is how mathematics is done in real life.
2. The write-up phase. This consists of writing your solutions once you have an idea of how the problem can be solved. You should do this entirely by yourself. Be alone when you write your solutions. If you collaborate on this part, or you copy part of your solutions from somebody else, or you have notes written by somebody else in front of you when you write your solu- tions, or you use a draft or sketch that you wrote in collaboration with somebody else, you are engaging in academic misconduct. The University of Toronto takes academic integrity very seriously. We are obligated to report all suspected instances of misconduct to OSAI. Please do not force us to do so. Every year, multiple students in this course are disciplined. For more in- formation, see https://www.artsci.utoronto.ca/newstudents/transition/academic/plagiarism.
One of the goals of having problem sets in this course is to improve your mathematical writing style. Your first assignment will concentrate on this task. Below are three advices on how to write up your solutions, each illustrated with examples. For each advice there is a poorly written solution by a fictitious student, lets name him John. You have to correct John's solutions and write the corrected version in the space provided.
1. Write your solutions in complete, grammatically correct sentences. Furthermore, make sure that words in your sentences agree with each other mathematically. For example, the phrase "the matrix A is linearly independent" does not mean anything - linear independence is a property of sets of vectors. Instead, one may write "the set of columns of A is linearly independent" or "{A} is linearly independent", the latter means A is a nonzero matrix (why?). Another example would be "{1, 2, -1} is 3"; what is, probably, meant is "the size of {1, 2, -1} is 3".
John was solving the following problem "Prove that the dimension of the column space of a matrix A is equal to the number of leading 1s in the reduced row echelon form of A." His answer was: "Let A be a matrix. The column space of A is the pivot columns of A (proven in class), and therefore is equal to the number of the leading 1s in the RREF of A.". Write the corrected solution here. (4 marks)
2. Make sure that all variables used in your solutions are introduced with words like "for any", "let", "assume", "there exists", "suppose", etc. For example, "f = g if and only if f (x) = g(x)" should be something like "Let f, g be in F (R), then f = g if and only if for all x ∈ R, f (x) = g(x)."
Another example: "A is consistent implies Ax = b" should be "Assume A is an m by n matrix corresponding to an always consistent system of linear equations. This means for all b ∈ Rm, there exists x ∈ Rn such that Ax = b."
John was trying to prove that F (R) satisfies axiom 8 of the definition of the (real) vector space, he wrote "Since (1f ) (x) = 1f (x) = f (x), we have 1f = f ." Write the corrected solution here.
3. Start with premises, end with conclusions. Explain the logical connections between non- trivial steps. For example, John was trying to prove that for any vectors v1, v2, Span ({v1}) ⊂ Span({v1, v2}). He wrote: "Let v1, v2 be vectors in a real vector space V . Span ({v1}) ⊂ Span ({v1, v2}). {v1} ⊂ {v1, v2} true." Instead, the solution should be "Let v1, v2 be vectors in a real vector space V . Since {v1} ⊂ {v1, v2}, it follows that Span({v1}) ⊂ Span({v1, v2}) from the theorem proven in class."
John also attempted to prove that for any vector v in a vector space V , 0v = 0. He wrote: "Let v be a vector in a vector space V , then 0v = 0, (0 - 0)v = 0, 0v - 0v = 0, 0v = 0v, true." Write the corrected solution here.