1. CHAPTER ONE Matrices and System Equations
Objective:To provide solvability conditions of a linear equation Ax=b and introduce the Gaussian elimination method, a systematical approach in solving Ax=b, to solve it.

2. Outline Motivative Example.
Elementary row operations and Elementary Matrices.
Some Basic Properties of Matrices.
Gaussian Elimination for solving Ax=b.
Solvability conditions for Ax=b.

3. Motivative Example (curve fitting)
Given three points( )( )( ),find a polynomial of degree 2 passing through the three given points.
Solution: Let the polynomial be
Where a,b and c are to be determined
Ax=b

4. Question: Why transform to matrix form?
To provide a systematic approach and to use computer resource.

5. Question: How to solve Ax=b systematically?
One way is to put Ax=b in triangular form,which
can be easily solved by back-substitution.
Definition: A system is said to be in triangular form if in the k-th equation the coefficients of thee first (k-1) variables are all zero and the coefficient of xk is nonzero ( k = 1,…,n)

6. Eg1:

7. Solution: By elementary row operations as described below.
Question: How to put Ax=b in triangular form while leaving the solution set invariant?
Solution: By elementary row operations as described
below.
Definition: Two systems of equations involing the same variables are said to be equivalent if they have the same solution set.

8. Before introducing elementary operation, we recall some definitions and notations.
(§ 1.3)
Equality of two matrices.
Multiplication of a matrix by a scalar.
Matrix addition.
Matrix multiplication.
Identity matrix.
Multiplicative inverse.
Nonsingular and singular matrix.
Transpose of a matrix.

9. Definitions Def. If and , then the where .
Matrix Multiplication ,
where
Def. An (n × n) matrix A is said to be nonsingular
or invertible if there exists a matrix B such that
AB=BA=I. The matrix B is said to be a
multiplicative inverse of A. And B is denoted
by A-1.
Warning: In general, AB≠BA. Matrix multiplication is not commutative.

10. Definitions (cont.)
Def. The transpose of an (m × n) matrix A is the (n ×
m) matrix B defined by for j=1,…,n and
i=1,…,m. The transpose of A is denoted by AT.
Def. An (n × n) matrix A is said to be symmetric if AT=A .

11. Some Matrix Properties
Let be scalars,A,B and C be matrices
with proper dimensions.
(Commutative Law)
(Associative Law)
(Distributive Law)

12. Some Matrix Properties (cont.)

13. Notations ,
The matrix is called an
augmented matrix.
In general, or

14. Moreover,we define

15. Def: Let and Then
is said to be a linear combination of
Note that We have the next result.
Theorem1.3.1: Ax=b is consistent b can be written
as a linear combination of colum vectors
of A.

16. Calories Burned Per Hour
Application 1: Weight Reduction
Table 1
Calories Burned Per Hour
Weight in lb
Exercise Activity
152
161
170
178
Walking 2 mph
213
225
237
249
Running 5.5 mph
651
688
726
764
Bicycling 5.5mph
304
321
338
356
Tennis
420
441
468
492

17. Hours Per Day For Each Activity
Application 1: Weight Reduction (cont.)
Table 2
Hours Per Day For Each Activity
Exercise schedule
walking
Running
Bicycling
Tennis
Monday
1.0
0.0
Tuesday
2.0
Wednesday
0.4
0.5
Thursday
Friday

18. Application 1: Weight Reduction (end)
Solution:

19. Production Costs Per Item (dollars)
Application 2: Production Costs
Table 3
Production Costs Per Item (dollars)
Product
Expenses A B C
Raw materials
0.1
0.3
0.15
Labor
0.4
0.25
Overhead and
miscellaneous
0.2

20. Amount Produced Per Quarter
Application 2: Production Costs (cont.)
Table 4
Amount Produced Per Quarter
Season
Product
Summer
Fall
Winter
Spring A
4000
4500 B
2000
2600
2400
2200 C
5800
6200
6000

21. Application 2: Weight Reduction (cont.)
Solution:

22. Application 2: Weight Reduction (cont.)
Solution:

23. Amount Produced Per Quarter
Application 2: Production Costs (end)
Solution:
Table 5
Amount Produced Per Quarter
Season
Summer
Fall
Winter
Spring
Year
Raw materials
1,870
2,160
2,070
1,960
8,060
Labor
3,450
3,940
3,810
3,580
14,780
Overhead and miscellaneous
1,670
1,900
1,830
1,740
7,140
Total production cost
6,990
8,000
7,710
7,280
29,980

24. Application 5: Networks and Graphs (P.57)

25. Application 5: Networks and Graphs (cont.)
DEF.

26. Application 5: Networks and Graphs (end)
Theorem
If A is an n × n adjacency matrix of a graph and represents
the ijth entry of Ak, then is equal to the number of walks of length
from to Vi to Vj.

27. Application 6: Information Retrieval (P.59)
Suppose that our database, consists of these book titles:
B1. Applied Linear Algebra
B2. Elementary Linear Algebra
B3. Elementary Linear Algebra with Applications
B4. Linear Algebra and Its Applications
B5. Linear Algebra with Applications
B6. Matrix Algebra with Applications
B7. Matrix Theory
The collection of key words is given by the following alphabetical
list：
algebra, application, elementary, linear, matrix, theory

28. Array Representation for Database of Linear Algebra Books
Application 6: Information Retrieval (cont.)
Table 8
Array Representation for
Database of Linear Algebra Books
Books
Key Words B1 B2 B3 B4 B5 B6 B7
algebra 1
application
elementary
linear
matrix
theory

29. Application 6: Information Retrieval (end)
If the words we are searching for are applied, linear, and algebra,
then the database matrix and search vector are given by
If we set y= ATx, then

30. Let’s back to solve Ax=b
Eg2

31. Three types of Elementary row operations.
(§ 1.2)
Three types of Elementary row operations.
I. Interchange two row.
II. Multiply a row by
III. Replace a row by its sum with a multiple of
another row.

32. Lead variables and free variables(p.15)
Eg:
, and are lead variables while and
are free variables.

33. Def. A matrix is said to be in row echelon form if
(i) The first nonzero entry in each row is 1.
(ii) If row k does not consist entirely of zero,
the number of leading zero entries in row
k+1 is grater then the number of leading
zero entries in row k.
(iii) If there are rows whose entries are all zero, they
are below the rows having nonzero entries.
Def. The process of using row operations I, II, and III to
transform a linear system into one whose augmented
matrix is in row echelon form is called Gaussian
elimination.

34. Overdetermined and Underdetermined
Def. A linear system is said to be overdetermined
if there are more equations(m) than unknowns
(n). (m > n)
Warning: Overdetermined systems are usually (but not always) in consistent.
Def. A system of m linear equations in n unknowns
is said to be underdetermined if there are
fewer equations. (m < n)

35. Reduced Row Echelon Form
Def. A matrix is said to be in reduced row echelon
form if:
(i) The matrix is in row echelon form.
(ii) The first nonzero entry in each row is the
only nonzero entry in its column.
Def. The process of using elementary row operations
to transform a matrix into reduced row echelon
form is called Gauss-Jordan reduction.

36. Application 2: Electrical Networks (P.22)

37. Application 2: Electrical Networks (end)
Kirchhoff’s Laws:
1. At every node the sum of the incoming currents equals the
sum of the outgoing currents.
2. Around every closed loop the algebraic sum of the voltage
must equal the algebraic sum of the voltage drops.

38. Application 4: Economic Models For Exchange of Goods (P.25)
F M C F M C
1/2
1/4
1/3
1/2
1/4

39. (§ 1.4) Elementary Matrices
Type I ( ): Obtained by interchanging rows i and j
from identity matrix.
Type II ( ): Obtained from identity matrix by
multiplying row i with
Type III ( ): Obtained from identity matrix by adding
to row j.

40. Elementary Row / Column Operation
means performing type I row operation on A.
means performing type II row operation on A.
means performing type III row operation on A.
means performing type I column operation on A.
means performing type II column operation on A.
means performing type III column operation on A.

41. Theorem1.4.2:
If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.
With
The solution set of a linear equations is invariant under three types row operation.
 and have the solution set.

42. Def. A matrix B is row equivalent to A if there exists a
Row Equivalent (P.71)
Def. A matrix B is row equivalent to A if there exists a
finite sequence of elementary matrices such that
Theorem1.4.3
(a) A is nonsingular.
(b) Ax=0 has only the trivial solution 0.
(c) A is row equivalent to I.

43. Proof of Theorem 1.4.3 (a) (b) Let be a solution of Ax=0. (b) (c)
Let A ~ U, where U is in reduced row echelon form.
Suppose U contains a zero row.
by Th1.2.1, Ux=0 has a nontrivial solution
thus A~I.
(c) (a)
A~I  A= E1 …… Ek for some E1 … Ek
∵ each Ei is nonsingular.
∴ A is nonsingular. (by Th.1.2.1)
row

44. Corollary1.4.4 Ax=b has a unique solution A is nonsingular.
Pf: " “ The unique solution is
" " Suppose is the unique solution and A is
singular.
is also a solution of Ax=b.
A is nonsingular.

45. BUT in general, and AB=AC B=C.
Eg.
Moreover,AC=AB while

46. Method For Computing = (I | Ek…E1‧I) ( by ) = (I | A-1)
If A is nonsingular and row equivalent to I, so
there exists elementary matrices such that
then,
Ek…E1(A | I)= (Ek…E1‧A | Ek…E1‧I) ( by )
= (I | Ek…E1‧I) ( by )
= (I | A-1)
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47. Example 4. (P.73) Sol: Q: Compute A-1 if .
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48. Example 4. (cont.) Sol: Q: Compute A-1 if .
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49. Diagonal and Triangular Matrices
Def. An n × n matrix A is said to be upper triangular if aij=0 for i > j and lower triangular if aij=0 for i > j.
Def. An n × n matrix B is diagonal if aij=0 whenever i ≠ j.
Triangular Factorization
If an n × n matrix C can be reduced to upper triangular form
using only row operation III, then C has an LU factorization.
The matrix L is unit lower triangular, and if i > j, then lij is the multiple of t he jth row subtracted from the ith row during the reduction process.
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50. Example 6. (P.74)
row operation III
Mark:
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51. Block Multiplication
Let A be an m × n matrix and B is an n × r matrix.
It is often useful to partition A and B and express the
product in terms of the submatrices of A and B.
In general, partition B into columns
then
partition A into rows , then
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52. Block Multiplication (cont.)
Case 1.
Case 2.
Case 3.
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53. Block Multiplication (cont.)
Case 4.
Let
then
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54. Example 2. (P.85) Let A be an n × n matrix of the form ,
where A11 is a k × k matrix (k < n ) .
Show that A is nonsingular if and only if A11 and A22
are nonsingular.
Solution:
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55. Scalar / Inner Product Give two vectors ,
This product is referred to as a scalar product or an
inner product.
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56. Outer Product Give two vectors ,
The product is referred to as the outer product of
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57. Outer Product Expansion
Suppose that , then
This representation is referred to as an outer product
expansion .
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