length of a vector linear algebra
Tài liệu Elements of abstract and linear algebra docx
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17/01/2014, 04:20
... four chapters, the linear algebra follows easily. Finishing the chapter on linear
algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6
continues the material to complete ... cosets of a subgroup H. In
Chapter 1
Background and Fundamentals of
Mathematics
This chapter is fundamental, not just for algebra, but for all fields related to mathe-
matics. The basic concepts are ... chapters can stand alone as a one semester course in abstract
algebra. However they are structured to provide the background for the chapter on
linear algebra. Chapter 2 is the most difficult part...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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15/12/2013, 04:15
... to trade@cup.cam.ac.uk (outside North America).
Chapter 2. Solution of Linear
Algebraic Equations
2.0 Introduction
A set of linear algebraic equations looks like this:
a
11
x
1
+ a
12
x
2
+ a
13
x
3
+ ... sophisticated black-box program packages. Several good ones
are available, thoughnot always in C. LINPACK was developed at Argonne National
Laboratories and deserves particular mention because it ... more of the M equations is a linear combination of
the others, a condition called row degeneracy, or if all equations contain certain
variables only in exactly the same linear combination, called...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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15/12/2013, 04:15
... America).
Elimination on Column-Augmented Matrices
Consider the linear matrix equation
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a
33
a
34
a
41
a
42
a
43
a
44
·
x
11
x
21
x
31
x
41
x
12
x
22
x
32
x
42
x
13
x
23
x
33
x
43
y
11
y
12
y
13
y
14
y
21
y
22
y
23
y
24
y
31
y
32
y
33
y
34
y
41
y
42
y
43
y
44
=
b
11
b
21
b
31
b
41
b
12
b
22
b
32
b
42
b
13
b
23
b
33
b
43
1000
0100
0010
0001
(2.1.1)
Here ... unchanged and in no way scrambled if we
replace any row in A by a linear combination of itself and any other row,
as long as we do the same linear combination of the rows of the b’s and 1
(which then ... The matrix
inverse of A is gradually built up in A as the original A is destroyed. Likewise,
the solution vectors x can gradually replace the right-hand side vectors b and share
the same storage,...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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15/12/2013, 04:15
... (North America only),or send email to trade@cup.cam.ac.uk (outside North America).
void qrsolv(float * *a, int n, float c[], float d[], float b[])
Solves the set of
n
linear equations A · x = b.
a[ 1 ... In many applications only the
part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv.
98
Chapter 2. Solution of Linear Algebraic Equations
Sample page from ... scratch, one can often update a matrix factorization in O(N
2
)
operations and use the new factorization to solve the next set of linear equations. The LU
decomposition is complicated to update...
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Tài liệu Determining the Length of Columns in a SQL Server Table doc
Ngày tải lên :
24/12/2013, 05:15
... Microsoft SQL
Server Books Online.
The GetSchemaTable( ) method of the DataReader also returns all column lengths. The
method returns a DataTable containing column metadata for a DataReader, ... = new DataSet( );
// Fill the schema and data.
da.FillSchema(ds, SchemaType.Mapped);
da.Fill(ds);
// Iterate over the table collection in the DataSet.
foreach(DataTable dt in ds.Tables) ... procedure takes one optional parameter. When this parameter
is the name of a table object, the result set returns data including the length, precision,
and scale of each column in columns named Length, ...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
Ngày tải lên :
24/12/2013, 12:16
... equations.
The advantage of Gaussian elimination and backsubstitutionover Gauss-Jordan
elimination is simply that the former is faster in raw operations count: The
innermost loops of Gauss-Jordan elimination, ... is
1
2
N
2
executions of a similar loop (one
multiplication plus one subtraction). For M N (only a few right-hand sides)
Gaussian elimination thus has about a factor three advantage over Gauss-Jordan.
(We ... section.
Gaussian elimination reduces a matrix not all the way to the identity matrix, but
only halfway, to a matrix whose components on the diagonal and above (say) remain
nontrivial. Let us...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
Ngày tải lên :
24/12/2013, 12:16
... β
22
β
23
β
24
00β
33
β
34
000β
44
=
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a
33
a
34
a
41
a
42
a
43
a
44
(2.3.2)
We can use a decomposition such as (2.3.1) to solve the linear set
A · x =(L·U)·x=L·(U·x)=b ... columns of
B instead of with the unit vectors that would give A s inverse. This saves a whole
matrix multiplication, and is also more accurate.
Determinant of a Matrix
The determinant of an LU ... Solution of Linear Algebraic Systems
(Engle-
wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18.
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear Equations
(New...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx
Ngày tải lên :
24/12/2013, 12:16
... Press).
2.4 Tridiagonal and Band Diagonal Systems
of Equations
The special case of a system of linear equations that is tridiagonal, that is, has
nonzero elements only on the diagonal plus or minus ... be
referenced by any manipulations and can have arbitrary values. Notice that the diagonal
of the original matrix appears in column m
1
+1, with subdiagonal elements to its left,
superdiagonal elements ... one that occurs
frequently. Also common are systems that are band diagonal,withnonzero elements
only along a few diagonal lines adjacent to the main diagonal (above and below).
For tridiagonal sets,...
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Tài liệu Solution of Linear Algebraic Equations part 12 pdf
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24/12/2013, 12:16
... c’s are scalars, but as matrix inversion if the a s and c’s are
themselves submatrices. Imagine doing the inversionof a very large matrix, of order
N =2
m
, recursively by partitions in half. At ... Wiley).
Winograd, S. 1971,
Linear Algebra and Its Applications
, vol. 4, pp. 381–388.
Pan, V. Ya. 1980,
SIAM Journal on Computing
, vol. 9, pp. 321–342.
Pan, V. 1984,
How to Multiply Matrices Faster
, ... fewer multiplication than in equation
(2.11.2), but many more additions and subtractions. It is not clear that anything
has been gained. But notice that in (2.11.3) the a s and b’s are never commuted.
Therefore...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx
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24/12/2013, 12:16
... or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).
We can define the norm of a matrix as the largest amplification of length that it is
able ... has an error term; but
we neglected the fact that the LU decomposition of A is itself not exact.
A different analytical approach starts with some matrix B
0
that is assumed to be an
approximate ... a linear
set than the precision of your computer’s floating-point word. Unfortunately, for
large sets of linear equations, it is not always easy to obtain precision equal to, or
even comparable...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
Ngày tải lên :
24/12/2013, 12:16
... b. (b) A singular matrix A maps a
vector space into one of lower dimensionality, here a plane into a line, called the “range” of A. The
“nullspace” of A is mappedto zero. The solutions of A · x ... throwing away one linear combination of the set of equations that
we are trying to solve. The resolution of the paradox is that we are throwing away
precisely a combination of equations that is ... d
range of A
d
c
(b)
(a)
A
x
b
c′
Figure 2.6.1. (a) A nonsingular matrix A maps a vector space into one of the same dimension. The
vector x is mapped into b,sothatxsatisfies the equation A · x =...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx
Ngày tải lên :
24/12/2013, 12:16
... NAG
library
[4]
has an analyze/factorize/operate capability. A substantial collection of
routines for sparse matrix calculation is also available from IMSL
[5]
as the Yale
Sparse Matrix Package
[6]
.
You ... a unit vector) .
The Sherman-Morrison formula can be directly applied to a class of sparse
problems. If you already have a fast way of calculating the inverse of A (e.g., a
tridiagonal matrix, ... the value of N
(namely 5) is ija[1]-2, and the length of each array is ija[ija[1]-1]-1, namely 11.
The diagonal element in row i is sa[i], and the off-diagonal elements in that row are in
sa[k]...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx
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21/01/2014, 18:20
... trade@cup.cam.ac.uk (outside North America).
2.8 Vandermonde Matrices and Toeplitz
Matrices
In Đ2.4 the case of a tridiagonal matrix was treated specially, because that
particular type of linear ... diagonal principal minors of the
original Toeplitz matrix vanish. (Compare with discussion of the tridiagonal algorithm in
Đ2.4.) If the algorithm fails, your matrix is not necessarily singular ... will notice that it is
specifically designed so that it takes on a value of zero at all x
i
with i = j, and has a value
of unity at x = x
j
.Inotherwords,
P
j
(x
i
)=δ
ij
=
N
k=1
A
jk
x
k−1
i
(2.8.4)
But...
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Tài liệu Solution of Linear Algebraic Equations part 10 docx
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21/01/2014, 18:20
... trade@cup.cam.ac.uk (outside North America).
2.8 Vandermonde Matrices and Toeplitz
Matrices
In Đ2.4 the case of a tridiagonal matrix was treated specially, because that
particular type of linear ... then, to have the factor L overwrite the
subdiagonal (lower triangular but not including the diagonal) part of A, preserving the input
upper triangular values of A. Only one extra vector of length ... i;
for(i=1;i<=n;i++) x[i]=(sa[i] != 0.0 ? b[i]/sa[i] : b[i]);
The matrix
A is the diagonal part of A, storedinthefirstnelements of sa.Sincethe
transpose matrix has the same diagonal, the flag itrnsp is not...
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Tài liệu Đề tài " Localization of modules for a semisimple Lie algebra in prime characteristic " pdf
Ngày tải lên :
16/02/2014, 06:20
... hand a choice of a Cartan subalgebra t⊆b defines an
4
An action of a Lie algebra
a
on a variety X is an action of
a
on O
X
by derivations.
Equivalently, it is a Lie algebra homomorphism from
a
to ... sufficiently large positive
characteristic. Next, one finds that for any smooth variety this algebra of
differential operators is an Azumaya algebra on the cotangent bundle. In the
case of the flag variety ... clearly G(k), and hence G
χ
(k) equivariant. Then our Azumaya
algebra is G
χ
(k) equivariantly identified with the pull-back of an Azumaya
algebra on χ
(1)
(see the proof of Theorem 5.1.1), and M
λ
χ
is...
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