1. 2.4 Linear Independence (線性獨立) and Linear Dependence(線性相依)
A linear combination(線性組合) of the vectors in V is any vector of the form c1v1 + c2v2 + … + ckvk where c1, c2, …, ck are arbitrary scalars.
A set of V of m-dimensional vectors is linearly independent(線性獨立) if the only linear combination of vectors in V that equals 0 is the trivial linear combination.
A set of V of m-dimensional vectors is linearly dependent (線性相依) if there is a nontrivial linear combination of vectors in V that adds up to 0.

2. Example 10: LD Set of Vectors (p.33)
Show that V = {[ 1 , 2 ] , [ 2 , 4 ]} is a linearly dependent set of vectors.
Solution
Since 2([ 1 , 2 ]) – 1([ 2 , 4 ]) = (0 0), there is a nontrivial linear combination with c1 =2 and c2 = -1 that yields 0. Thus V is a linear dependent set of vectors.

3. Linear dependent 線性相依 (p.33)
What does it mean for a set of vectors to linearly dependent?
A set of vectors is linearly dependent only if some vector in V can be written as a nontrivial linear combination of other vectors in V.
If a set of vectors in V are linearly dependent, the vectors in V are, in some way, NOT all “different” vectors.
By “different” we mean that the direction specified by any vector in V cannot be expressed by adding together multiples of other vectors in V. For example, in two dimensions, two linearly dependent vectors lie on the same line.

4. The Rank of a Matrix (p.34)
Let A be any m x n matrix, and denote the rows of A by r1, r2, …, rm. Define R = {r1, r2, …, rm}.
The rank(秩) of A is the number of vectors in the largest linearly independent subset of R.
To find the rank of matrix A, apply the Gauss-Jordan method to matrix A. (Example 14, p.35)
Let A’ be the final result. It can be shown that the rank of A’ = rank of A. The rank of A’ = the number of nonzero rows in A’. Therefore, the rank A = rank A’ = number of nonzero rows in A’.

5. Whether a Set of Vectors Is Linear Independent
A method of determining whether a set of vectors V = {v1, v2, …, vm} is linearly dependent is to form a matrix A whose ith row is vi. (p.35)
- If the rank A = m, then V is a linearly independent set of vectors.
- If the rank A < m, then V is a linearly dependent set of vectors.

6. 2.5 The Inverse of a Matrix (反矩陣) (p.36)
A square matrix(方陣) is any matrix that has an equal number of rows and columns.
The diagonal elements (對角元素)of a square matrix are those elements aij such that i=j.
A square matrix for which all diagonal elements are equal to 1 and all non-diagonal elements are equal to 0 is called an identity matrix(單位矩陣). An identity matrix is written as Im.

7. The Inverse of a Matrix (continued)
For any given m x m matrix A, the m x m matrix B is the inverse of A if BA=AB=Im.
Some square matrices do not have inverses. If there does exist an m x m matrix B that satisfies BA=AB=Im, then we write B= (p.39)
The Gauss-Jordan Method for inverting an m x m Matrix A is
Step 1 Write down the m x 2m matrix A|Im
Step 2 Use EROs to transform A|Im into Im|B. This will be possible only if rank A=m. If rank A<m, then A has no inverse (p.40)

8. Matrix inverses can be used to solve linear systems. (example, p.40)
The Excel command =MINVERSE makes it easy to invert a matrix.
Enter the matrix into cells B1:D3 and select the output range (B5:D7 was chosen) where you want A-1 computed.
In the upper left-hand corner of the output range (cell B5), enter the formula = MINVERSE(B1:D3)
Press Control-Shift-Enter and A-1 is computed in the output range

9. Inverse Matrix (反矩陣)

10. 反矩陣之性質
 p.41 #8a
 p.42 #9
 p.42 #10

11.  p.41 problems group 8a

12. Example :
例題：

13. Example (continued)
Exercise： problem 2, p.41, 5min

14. Orthogonal matrix (正規矩陣)
A-1=AT
Det(A)=1 or Det(A)=-1
Determine matrix A is an orthogonal matrix or not .
p.42 #11

15. Homogeneous System Equations
AX=b為一線性方程組,若b1=b2=…=bm=0，則稱為齊次方程組(homogeneous) ,以 AX=0表之。
x1=x2=…=xn=0為其中一組解，稱為必然解(trivial solution)
若x1、x2、…xn不全為0，則稱為非必然解(nontrivial solution)

17. Example:

18. Example:

20. Example : Solving linear system by inverse matrix

24. Exercise : Solving linear system by inverse matrix
(8 min)
Answer:

25. 2.6 Determinants(行列式) (p.42)
Associated with any square matrix A is a number called the determinant of A (often abbreviated as det A or |A|).
If A is an m x m matrix, then for any values of i and j, the ijth minor of A (written Aij) is the (m - 1) x (m - 1) submatrix of A obtained by deleting row i and column j of A.
Determinants can be used to invert square matrices and to solve linear equation systems.

26. Determinants(行列式)

27. Minor(子行列式) & Cofactor(餘因子)

28. 餘因子展開式

29. 行列式之性質(1)

30. 行列式之性質(2)

31. 行列式之性質(3)

32. 行列式之性質(4)

33. Adjoint matrix (伴隨矩陣)

34. Exercise :
adj A = ？
det(A) = ？
Answer：
det(A) = -34

35. Exercise :
若det(λI3-A)=0, λ=？
λ =-5 或λ=0或λ=3

36. 伴隨矩陣之性質

37. 行列式之性質

38. Cramer’s Rule

39. Cramer’s Rule 注意事項 若det(A) ≠0，則n元一次方程組為相容方程組， 其唯一解為
若det(A) =det(A1) =det(A2)= …=det(An) =0， 則n元一次方程組為相依方程組，其有無限多組解
若det(A) =0，而det(A1) ≠0 或det(A2) ≠0 ，…， 或det(An) ≠0，則n元一次方程組為矛盾方程組，其 為無解。

40. Example : Solving Linear System
The system has an infinite number of solutions

41. Exercise : Solving linear system by Cramer’s rule
Answer:

42. LU-Decompositions
Factoring the coefficient matrix into a product of a lower and upper triangular matrices.
This method is well suited for computers and is the basis for many practical computer programs.

43. Solving linear systems by factoring
Let Ax = b，and coefficient matrix A can be factored into a product of n × n matrices as A = LU， where L is lower triangular and U is upper triangular, then the system Ax = b can be solved by as follows：
Step 1. Rewrite Ax = b as LUx = b (1)
Step 2. Define a n × 1 new matrix y by Ux = y (2)
Step 3. Use (2) to rewrite (1) as Ly = b and solve this system for y.
Step 4. Substitude y in(2) and solve for x

44. Example : Solving linear system by LU decomposition
Step 1. A=LU, Ax=b  LUx=b

45. Step 2. Ux=y
Step 3. Ly=b
Solving y by forward-substitution.
y1=1, y2=5, y3=2

46. y1=1, y2=5, y3=2 Step 4. Solving y by backward-substitution.
x1=2, x2=-1, x3=2

47. Doolittle method =
Assume that A has a Doolittle factorization A = LU. L：unit lower-△，U：upper-△ =
The solution X to the linear system AX = b  , is found in three steps:       1.  Construct the matrices  L and U, if possible.       2.  Solve LY = b  for  Y  using forward substitution.     3.  Solve UX = Y   for  X  using back substitution. 

48. Crout method
Assume that A has a Doolittle factorization A = LU. L：lower-△，U：unit upper-△  =
The solution X to the linear system AX = b  , is found in three steps:       1.  Construct the matrices  L and U, if possible.       2.  Solve LY = b  for  Y  using forward substitution.     3.  Solve UX = Y   for  X  using back substitution. 

49. Solving Linear Equations (see *.doc)
Gauss backward-substitution 高斯後代法
-- forward-substitution
Gauss-Jordan Elimination 高斯約旦消去法 -- reduced row echelon matrix
Inversed matrix ─ solve Ax=b as x=A-1b
LU method ─ solve Ax=b as LUx=b
Cramer's Rule ─ x1=det(A1)/det(A),….