Matrices and Linear Systems

Transcription

Matrices and Linear Systems



Roughly speaking, matrix is a rectangle array
We shall discuss existence and uniqueness of
solution for a system of linear equation.
The method of Gauss ellimination will be given to
solve the system .
Advanced Engineering Mathematics by
Erwin Kreyszig
Page 274 (1)
Erwin Kreyszig
Page 274 (2)
Erwin Kreyszig
Pages 274-275
Erwin Kreyszig
Page 275 (1)
Erwin Kreyszig
Page 275 (2)
Erwin Kreyszig
Page 276 (1)
Erwin Kreyszig
Page 276 (2)
Erwin Kreyszig
Page 279a
Continued
Erwin Kreyszig
Page 279b
Erwin Kreyszig
Page 280 (1)
Erwin Kreyszig
Page 280 (2)
Erwin Kreyszig
Page 282
Erwin Kreyszig
Page 283 (1)
Erwin Kreyszig
Page 283 (2)
Erwin Kreyszig
Page 283 (3)
Erwin Kreyszig
Page 284 (1)
Erwin Kreyszig
Page 284 (2)
Erwin Kreyszig
Page 288 (1)
Erwin Kreyszig
Page 288 (2a)
Continued
Erwin Kreyszig
Page 288 (2b)
Erwin Kreyszig
Page 291a
Continued
Erwin Kreyszig
Page 291b
Continued
Erwin Kreyszig
Page 291c
Erwin Kreyszig
Page 292 (1)
Erwin Kreyszig
Page 292 (3)
Erwin Kreyszig
Page 294a
the row reduced echelon form
1 2 / 3 1 / 3
1 0 1 
0
  0 1  1
1

1




0
0 0 0 
0
0 
Advanced Engineering Mathematics by Erwin Kreyszig
Copyright  2007 John Wiley & Sons, Inc. All rights reserved.
1 3 0 5  1
Example : 
is a row reduced echolon matrix.

0 0 1 0 2 
1. Find row reduced echelon forms of
 1 4
0 0

 0 0
2. Show
1 1
 2 1 3
0 0 ,  0 1 1
 2 0 1
0 1
that they are inconsistent systems.
2x 1   3 x 2
1
2x 1  3 x 2
6
x1 
3x 2
0
4x 1  6x 2
 18
x1 
4x 2
3
Erwin Kreyszig
Page 294b
Continued
Erwin Kreyszig
Page 294c
Erwin Kreyszig
Vector Spaces

A quantity such as work, area or energy which is
described in terms of magnitude alone is called a scalar.

A quantity which has both magnitude and direction for
its describtion is called a vector.
A vector is an element of vector space.
Erwin Kreyszig
Definiton: A vector space V in R is the set satisfying
1.If u , v  V and a  R, then u  v, au  V .
2. u  v  v  u
3. u  (v  w)  (u  v)  w
4. u  0  u (a unique zero element)
5. u  (u )  0 (a uique additive inverse)
6. (a  b)u  au  bu
7.(ab)u  a(bu )
8.1u  u (1 is identity scalar)
9.a(u  v)  au  av
Erwin Kreyszig
Examples for vector spaces
1. V  {0}
2. R n
3. R mxn (space of matrices)
4. Pn [x] (space of polynomials)
5. F[a, b] (function - space on [a, b])
n
6. C [a, b] (continously differentiable func.)
Erwin Kreyszig
Linear Dependence
Define a linear combinatio n of nonzero vectors
c1u1  c 2 u 2  ...  c n u n ,
where u i  V , ci  R, i  1,2,3,...
If the equation c1u1  c 2 u 2  ...c n u n  0
has only trivial solution for all ci , then
u1 , u 2 ,...u n are linealy independent.Otherwise,
if any ci is zero then they are linearly dependent.
Erwin Kreyszig
Page 297 (2)
Erwin Kreyszig
Page 298 (2)
Rank of A is 2 because the first two rows are linearly independent.
Erwin Kreyszig
Example
 2 1 3
1 0 0
A   0 1 1 is row equivalent to 0 1 0.
 2 0 1
0 0 1
rankA  3
Erwin Kreyszig
Page 298 (3)
Erwin Kreyszig
Page 299
Erwin Kreyszig
Dimension of a vector space V




SpanS= All linear combinations of vectors of the subset
S of V .
A basis for V is a linearly independent subset S of V
which spans the space V.
That is, SpanS= V where S is lin. İndep.
dimV= The number of vectors in any basis for V.
V is finite-dimensional if V has a basis consisting of a
finite number of vectors.
Erwin Kreyszig
Note: (6) is known as dimension theorem
Pages 302-303a
Continued
Erwin Kreyszig
Pages 302-303b
Erwin Kreyszig
Page 304
Erwin Kreyszig
Page 305 (1)
Erwin Kreyszig
Page 305 (2)
Erwin Kreyszig
Page 305 (3)
Erwin Kreyszig
Example
3x1  2 x 2  3x3  2 x 4  1
x1  x 2  x3  0 x 4  3
x1  2 x 2  x3  x 4  2
x0  1 2 0 3 , x h  c 1 0 1 0
T
Erwin Kreyszig
T
Determinant
Determinant is a function form square
matrices to scalars.
Our efficient computational procedure will be cofactor
expansion.
Erwin Kreyszig
Page 306a
Continued
Erwin Kreyszig
Page 306b
Erwin Kreyszig
Page 307a
Continued
Erwin Kreyszig
Page 307b
Erwin Kreyszig
Page 308a
Continued
Erwin Kreyszig
Page 308b
Erwin Kreyszig
Page 309 (1)
Erwin Kreyszig
Page 309 (2a)
Continued
Erwin Kreyszig
Page 309 (2b)
Erwin Kreyszig
Page 310
Erwin Kreyszig
Page 311 (1)
Erwin Kreyszig
Page 311 (2)
Erwin Kreyszig
Page 312a
Continued
Erwin Kreyszig
Page 316
Erwin Kreyszig
Pages 317-318a
Continued
Erwin Kreyszig
Pages 317-318b
Continued
Erwin Kreyszig
Pages 317-318c
Erwin Kreyszig
Page 318
Erwin Kreyszig
Page 319a
Continued
Erwin Kreyszig
Page 319b
Erwin Kreyszig
Page 320 (1)
Erwin Kreyszig
Page 320 (2)
Erwin Kreyszig
Page 321 (1)
Erwin Kreyszig
Page 321 (2)
Erwin Kreyszig
Page 322
Erwin Kreyszig
Erwin Kreyszig
Examples
Erwin Kreyszig
Linear Transformations
Examples: Zero transform, identity operator, scalar-multiple
operator,reflection , projection , rotation, differential
transform, integral transform.
Erwin Kreyszig
Representiation Matrix
Erwin Kreyszig
Example
Erwin Kreyszig
Example:
Find the representiation matrix of
Erwin Kreyszig
Range and Null (Kernel) spaces
Let F : V  W be a linear tra nsform.
NullF  {u : F (u )  0, u  V }.
RangeF  {v : v  F (u ), u  V }  W
includes all images vectors.
dim (NullF)  nullityF
dim (RangeF)  rankF
Theorem : rankF  nullityF  dim V.
Erwin Kreyszig
Pages 331-332a
Continued
Erwin Kreyszig
Pages 331-332b
Continued
Erwin Kreyszig
Pages 331-332c
Continued
Erwin Kreyszig
Pages 331-332c
Erwin Kreyszig

Matrices and Linear Systems

Transcription

Similar documents

Der Tagesspiegel Nr. 17597 VOM 23.11.2001 SEITE 030