Summary
Summary Linear Algebra (JBM075)
- Course
- Institution
- Book
Summary with examples, exercises and some proofs
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
MAT1503 Assignment 5 (DETAILED ANSWERS) 2024 - DISTINCTION GUARANTEED
Seller
Follow
tnmsterk
Content preview
,A echelon formsystem of the
is matrix has
consisenting row
an no
the
gorn
:
of
10 ..... 06
· where b + 0
A thus consisent in case has least solution
system is it at one
·
, .
o
unique
solution When there are variables Otherwise infinite
no
gree ,
amount solutions
of
Vectors
A matrix with just one column is called a rector
m= (3) ,
v =
(2)
Note that n + U
.
The complete
Jet vectors with two entrice is denoted by R
of "
, rectors
With three entries
by ,
and rectors With n entrie by R!
=)
Definitions
Let h and v be vectors in A :
= no v .
Their u
n + v is then defined a : n + v =
W
Definition :
Let
h be a vector in R and let Ch be a salar .
Then , the
(en)
ecalar multiplication with
of
> h is given by :
2 =
The o vector ,
is a vector whose entries are all
equal
to 0
.
Then , the
following propertie hold for all vectore U, v
,
WER and
= calcr <, d = k + v = V +4
·
(u + v) +w = n + (V + w)
· uto = 0 +u = u
4 + u
( 4)
-
· n + -
= =
0
,·
c(u+ u) = cu + 2r
·
(( + d)u = Eu + du
·
<(du) = (d) v
·
In = u
Given are the rectors v
,, We
, ...,
Un &V and Scalers C
,, 12 , ..., In
.
Y =
C , Vi + C2Ve +.... + CUn
is called a linear combination
of u
, ....,
Un with
treights < ,
...,
<
Span
If then the all lineur combination
"
V
,
...
un are in ,
cet of of
V
, ..., Un
is denoted
by Span &U, .....) Vul and termed the subset
of B
spained by V
1
, ... m
Thus :
SpanEU
, ..... rub = 2, v , + Seve +.... + CrVK
Span(s) =
[ civ : 1 vies]
The recrorb is in spanEU ,, Ve , ...,
U. 3
if : X, V , +
X2 Vn +... + X - Un = b
has a solution .
Thus the zero , rector is always in the span
.
Span 327 hae a line through the origin .
Spand U
,, Wz3 is a plane through the origin
Matrices and rectors
#f A is a mx n matrix ,
and B is a 1x matrix , then product C= AB
is an MXr matrix
. The (i)j) entry of a product is computed an
follows :
Cij = Giz bij + diz bizj + - ... + Gin
buj.
Let A be an Mx1 matrix
. Then , the following statements are
either all true or all false :
& For each b in
M
the equation Ax = b has a volution
,
& Each b in Ah is linear combination of the columns
a
ofA
② The column of
A opan Ru
& A has a pivot position in
every row
.
, Werke 2:
Homogeneous lineur systems
Homogeneous if it can be written an Ax = 0 where o is the zero rector
,
in B
Theorem :
For a
homogenous system with evariables and linear equations , there are
infinitely maysolutions ig m =1
Thus ,
in case there is at least one
free variable ,
then the cyclem
has infinitely many solutions .
#f X1 = -
X g
,
Xz = X 3 and X 3 is
free
The general of Ax = o has the vector solution :
X =
() [) =
=
= (i)
Thus X = X3 v
(di& S () (0) x(i)
X, = 3 -
xy
+
= 1 + xxx
= =
x = -
O O g Xs is free
The X =
p + X 3
Linear independente
Definition :
A V Un linearly independent inf the colution
setogrectors Va Vz is
"
,, , < ,
to the
vector equation X , V1 + X Vz +... + XeUn = 0
hue X1 X2 o solution
Only Xn = as a
, , ....,
Thurs ,
we can also make the definition for linear dependence :
Definition :
A get
of Vector
v
, Ve , ...,
Un is
linearly dependent if there are
Backers XI < +1 ,
-
, with at least one salar not zero ,
such that :
X vi + X2Vz + -. . + Xn Un = 0
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller tnmsterk. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.62. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
61001 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now