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
|
![Class Photo](/images/thumb/6/6b/14-240-ClassPhoto.jpg/310px-14-240-ClassPhoto.jpg) Add your name / see who's in!
|
|
|
Boris
Theorem
Let
be a
matrix and
be the matrix
with two rows interchanged. Then
. Boris decided to prove the following lemma first:
Lemma 1
Let
be a
matrix and
be the matrix
with two adjacent rows interchanged. Then
.
All we need to show is that
. Assume that
is the matrix
with row
of
interchanged with row
of
. Since the determinant of a matrix with two identical rows is
, then:
![{\displaystyle det(A)+det(B)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6f221712ba4ef348447412c519a2ea9da0a699)
![{\displaystyle det(A)+det(B)=det{\begin{pmatrix}...\\A_{i}\\A_{i+1}\\...\end{pmatrix}}+det{\begin{pmatrix}...\\A_{i+1}\\A_{i}\\...\end{pmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4b54b5ed78acb8cb907fb8abe4ac3ac04e5053)
.
Since the determinant is linear in each row, then we continue where we left off:
![{\displaystyle det(A)+det(B)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6f221712ba4ef348447412c519a2ea9da0a699)
![{\displaystyle det{\begin{pmatrix}...\\A_{i}\\A_{i}\\...\end{pmatrix}}+det{\begin{pmatrix}...\\A_{i}\\A_{i+1}\\...\end{pmatrix}}+det{\begin{pmatrix}...\\A_{i+1}\\A_{i+1}\\...\end{pmatrix}}+det{\begin{pmatrix}...\\A_{i+1}\\A_{i}\\...\end{pmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d78436acae9ef478a4fff58a8fb22b8a63a6190)
.
Then
and
.
Nikita