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Elementary Linear Algebra
A Matrix Approach
Sheng-Lung Huang ()Office: Room 348, EE-II BuildingTel:
02-33663700 ext. 348Email: [email protected]
3/2007
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Syllabus
1. Matrices, Vectors, and Systems of Linear Equations2. Matrices
and Linear Transformations3. Determinants4. Subspaces and their
Properties5. Eigenvalues, Eigenvectors, and Diagonalization6.
Orthogonality7. Vector Spaces
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Algebra
Algebra A branch of mathematics in which mathematical relations
are explored by using letters or symbols to represent numbers.
Linear Algebra The study of vectors, linear transformations,
systems of linear equations, and vector spaces (also called linear
spaces), in finite dimensions. Linear algebra has extensive
applications in the natural sciences and the social sciences, since
nonlinear models can often be approximated by a linear model.
Abstract algebra The study of algebraic structures such as
groups, rings, fields, and vector spaces.
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Chapter 1.1 Matrices and Vectors
DefinitionsSize Square matrixEntryEqualTraceTransposeSumScalar
multiple
2. Matrix arithmetic and operation
Component3.Subtraction
Vector
Rn
Zero matrixSub matrix
1. Matrix
,
p. 2-6
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MATLAB in NTU
: [[email protected]] : 2006927: [email protected]: matlab
33665015
matlabMATLABSIMULINKSymbolic_Toolbox(103)117312(52)
MATLAB
MATLABSIMULINKSymbolic_ToolboxNT$6200.-()
NT$15000-38000.-10NT$6000-15000.-25NT$3000-9500.-
http://oper.cc.ntu.edu.tw/
email [email protected]
2006/9/27
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The Geometry of Vectors
2D
Fig. 1.1, 1.2
3D
Fig. 1.5 p. 7-9
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Chapter 1.2 Linear Combinations, Matrix-Vector Products, and
Special Matrices
Definitions1. Linear combination
2. Identity matrix
3. Rotation matrix
4. Matrix-vector product
=
cossinsincos
A
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Chapter 1.3 Systems of Linear Equations
Elementary row operation
1. Interchange any two rows of the matrix Interchange
operation
2. Multiply every entry of some row of the matrix by the same
nonzero constantAdd a multiple of one row of the matrix to another
row
Scaling operation
3. Row additionoperation
Note:1. Every elementary row operation can be reversed.2.
Perform elementary operation on the augmented matrix
will not affect its solutions.p. 27
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Row Echelon Form
p. 28
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Chapter 1.4 Gaussian Elimination
Johann Carl Friedrich Gauss, Brunswick, Germany: 1777-1855
Gauss law
Gaussian beam
Gaussian distribution
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Gaussian Elimination
=
716560601131110
5254200A
The most efficient algorithm toobtain the reduced rowechlon
form.
=1210000
06021002101010
R
m x n = 3 x 7 Elementaryrow operations
Reduced row echelon form
-- Pivot positions: The positions that contain the leading
entries of thenonzero rows of R.
-- Pivot column: A column of A that contains some pivot position
of A.-- Rank: Number of nonzero rows in R. i.e. # of nonredundant
eqs.-- Nullity: n - rank A p. 36, 42
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Gaussian Elimination Procedure
Step 1 Determin the leftmost nonzero column. This is a pivot
column and, the topmost position in this column is a pivot
position.
Step 2 In the pivot column, choose any nonzero entry in a row
that is not ignored, and perform row interchange to bring this
entry into the pivot position.
Step 3 Add a multiple of the row containing the pivot position
to each lower row to change each entry below the pivot posotion
into zero.
Step 4
Ignore all the rows that contain previous pivot positions. If
every row of the matrix has been ignored, or if the rows that are
not ignored contain only zero entries, begin Step 5 using the last
nonzero row of the matrix. Otherwise, repeat Steps 1-4 on the
submatrix consisting of the rows that are not ignored.
Step 5If the leading entry of the row is not 1, perform the
scaling operation to make it 1. Then add a multiple of this row to
every preceding row to change each entry above the pivot position
into zero.
Step 6 If Step 5 was performed using the first row, stop.
Otherwise repeat Step 5on the preceding row.p. 36-40
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Gaussian Elimination Procedure
xxxxxxxnzxxxxnz
00
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00
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxnz
xxxxxxnz
10001000
xxxxxnzxxxxnz
1000
xxxxxxnzxxxxnz
000
xxxxnzxxxnz
100000
xxxxxxxnz
1000100
xxxxxx
100010001
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Example 1
=
716560601131110
5254200A
=
7165606052542001131110
A
=
716560601131110
5254200A
=
1210000012042002501110
A
=
242000052542001131110
A
=1210000
06021002101010
A
=
1310131260052542001131110
A
=
1310131260052542001131110
A
Number of arithmetic operations needed ~ n3
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Theorem 1.4 Test for Consistency
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Proof of Theorem 1.4
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Chapter 1.5* Applications of Systems of Linear Equations
All stared sections will be skipped exceptsection 6.6 (Singular
Value Decomposition).
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Chapter 1.6 The Span of a Set of Vectors
Span
Example 1
p. 60
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Span, linear combination, and system of linear eq.
p. 62
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Theorem 1.5
p. 64
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Proof of Theorm 1.5
(b) (c)
p. 64
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Theorem 1.6
p. 65
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Proof of Theorem 1.6
p. 65
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Chapter 1.7 Linear Dependence and Linear Independence
p. 68-69
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Examples
Example 1
Example 2
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Theorem 1.7
p. 71-72
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Proof of Theorem 1.7
p. 72
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Theorem 1.8
p. 73
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Proof of Theorem 1.8
p. 73-74
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Properties of Linear Depend. and Independ Sets
p. 74
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Theorem 1.9
p. 75
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Proof of Theorem 1.9
p. 75-76
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Summary
Theorem 1.5 (page 64) Theorem 1.7 (page 71)
For For
The following statements about an m x n matrix A are
equivalent.
(a) The span of the columns of A is Rm. (a) The columns of A are
linearly independent.
(b) The equation Ax=b has at least one solution for each bin
Rm.
(b) The equation Ax=b has at most one solution for each bin
Rm.
(c) The rank of A is m. (c) The nullity of A is zero.
(d) The reduced row echelon form of A has no zero rows. (d) The
rank of A is n.
(e) The columns of the reduced row echelon form of A are
distinct standard vectors in Rm.
(f) The only solution to Ax=0 is 0.
nm nm
Elementary Linear AlgebraSyllabusAlgebraChapter 1.1 Matrices and
VectorsMATLAB in NTUThe Geometry of VectorsChapter 1.2 Linear
Combinations, Matrix-Vector Products, and Special MatricesChapter
1.3 Systems of Linear EquationsRow Echelon FormChapter 1.4 Gaussian
EliminationGaussian EliminationGaussian Elimination
ProcedureGaussian Elimination ProcedureExample 1Theorem 1.4 Test
for ConsistencyProof of Theorem 1.4Chapter 1.5* Applications of
Systems of Linear EquationsChapter 1.6 The Span of a Set of
VectorsSpan, linear combination, and system of linear eq.Theorem
1.5Proof of Theorm 1.5Theorem 1.6Proof of Theorem 1.6Chapter 1.7
Linear Dependence and Linear IndependenceExamplesTheorem 1.7Proof
of Theorem 1.7Theorem 1.8Proof of Theorem 1.8Properties of Linear
Depend. and Independ SetsTheorem 1.9Proof of Theorem 1.9Summary
