Notes on linear algebra (Monday 17th October, 2016, 23:10) page 105
We have now proven (87) in both Cases 1 and 2. Thus, (87) is proven]
Now, Proposition 2.19 (a) shows that
(ABlirj= Ai,lBl,j + Ai,232
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 115
P = H, x = I! and y = 74') yiEIdS Eu,u,n,nEu,a,n,n = 5H,HEH,H,H,H* Since Eu,a,n,n = Esau:
this rewrites as Burial-3, = 5, Eu! 2 1&1!
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 113
(since matrices are scaled entry by entry).
We have
sf; C = (In + (A 1) EM) C = INC + (/1 1) EMC
Kr"
= Hanear =C
= C + (A 1) EMC.
He
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 107
(according to the denition of k-lower-triangular).
But It is a positive integer. Thus. 1 g k, so that j+ 1 g jI k for every
Etc
j E
Notes on linear algebra (Monday 17th October. 2016. 23:10) page 106
conclude that (A1142 - - +13) Ar+1 is an (E + 1)-lower-triangu]ar a x a-matrix. Since
(A1712 - - - Ag) Ag+1 = A1712 + . -Ag+1. this
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 112
Proposition 3.90. Let a E N and m E IN. Let a E cfw_1,2,.,a. Let A be a
number. Let C be an m x n-matrix. Then, C53 is the m x a-mat
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 109
We can now prove Theorem 3.67 again:
Second proof of Theorem 3.67. Proposition 3.82 (b) shows that A is lower-unitriangular
if and o
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 111
Proposition 3.87 can be rewritten as follows: The matrix S 3 (where a E IN, where
a 6 cfw_1,2, . . ., a, and where A is a number) is
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 98
matrices (and upper addition matrices instead of lower addition matrice555). The
proof of this analogue would then proceed similarly
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 114
In other words, the i-th row of the a x m-matrix SC equals the i-th row of C. This
proves Claim 2.
Now, we have proven both Claim 1
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 99
Example 3.70. Let A be a strictly lower-triangular 4 X 4-matrix. Thus, A has the
0 0 0 0
a 0 0 0
form 19 c 0 0 for some numbers a, b,
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 101
We state some simple facts (which should have been clear from the example
already):
Proposition 3.73. Let a E IN. Let A be an a x a-
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 108
Proposition 3.33. Let n E ]N. Let A be an n X n-matrix such that A" = 0M.
Then, the matrix In A is invertible, and its inverse is (I
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 100
Example 3.72. Visually speaking, a square matrix A is k-lower-triangular if and
only if its nonzero entries begin no earlier than k
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 103
Now. we can easily see that
AIL = 0 whenever i < j + 1
53. But this means precisely that A is 1-lower-triangular (because this is ho
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 116
Denition 3.93. Let a E ]N. A scaling a x nmatrix means a matrix of the form S,
where A is a nonzero number, and Where a is an elemen
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 104
Now, Proposition 3.73 (d) is proven (since both its => and its 4: directions are
proven). D
Next, we state a fact which is crucial f
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 102
two elements 1" and j of cfw_1,2,. . .,n never satisfy 1' < j | k. Thus, the statement
(80) is vacuously true, and therefore true. I
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 117
(b) The invertibly lower-triangular 1 x 1-matrix ( 5 ) is a product of scaling
matrices and lower addition matrices: Namely, it is 8
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 110
3.13. The Anscaling matrices 83
Now we shall explore another kind of square matrices not unlike the matrices Ail.)
from Denition 3.5
Project1
Due,Tuesday,November27,2012
Purpose:
The purpose of this project is to develop a creative a short skit containing dialog between two
charactersusing
xtranormal
.Thedialogshouldsatisfythefollo
I must say that I was very intrigued by the love lab excerpt. I think that it would be an interesting
and perhaps rewarding experience to attend the love lab with my wife. We have only been married fo
Leave Claim # 11512492
3913136”)
Family Medical Leave Act (FMLA) Certification for
Employee’s Serious Health Condition1
Return completed form to: Aetna Life Insurance Company
PO Box 14560
Lexington, K