14-240/Tutorial-October14

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday 2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf 3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf 4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf 5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf 6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial 7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday 8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial 9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial 10 Nov 10 HW8, Monday, Tutorial 11 Nov 17 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Boris

Elementary and (Not So Elementary) Errors in Homework

(1) Let $M_{1}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},M_{2}={\begin{pmatrix}0&0\\0&1\\\end{pmatrix}},M_{3}={\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$ be matrices.

We want to equate $span(M_{1},M_{2},M_{3})$ to the set of all symmetric $2\times 2$ matrices. Here is the wrong way to write this:

$span(M_{1},M_{2},M_{3})={\begin{pmatrix}a&b\\b&c\\\end{pmatrix}}$.

Firstly, $span(M_{1},M_{2},M_{2})$ is the set of all linear combinations of $M_{1},M_{2},M_{3}$. To equate it to a single symmetric $2\times 2$ matrix makes no sense. Secondly, the elements $a,b,c,d$ are undefined. What are they suppose to represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those issues.

$span(M_{1},M_{2},M_{3})=\{{\begin{pmatrix}a&b\\b&c\\\end{pmatrix}}:a,b,c\in F\}$ where $F$ is an arbitrary field.