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Chapter15.7~16.7
15.7 平面向量
★向量(Vectors) 幾何意義:為有一定方向和長度的指向線段,向量上的箭頭為所指方 向。PS.若一個向量平行移動與另依線段完全重疊時,則此兩向量相 等。 定義 平面中的向量可用 2×1 的矩陣表示u : 稱為 u 的分量。將平面向量稱簡稱為向量。 PS.兩向量若其分量皆相等,則此兩向量相等。 ★向量的長度或大小—畢氏定理 向量u , 的長度: , 其中 , 是實數,

若一有向線段

,

;

,

的長度:

★利用向量求面積   x1  1 三角形面積= det   x 2  2  x  3 y1 1   y 2 1   y 3 1   y1 1   y 2 1   y 3 1   ;c 為一純量 ⟹ 將其分量相加 ,

  x1  平行四邊形面積= det   x 2   x  3 ★向量的運算 設u , 加法:u 減法:u v v ;v , u v ,

乘法:cu c .c ⟹ 由純量乘以分量得之 ★兩向量的夾角 設有兩向量 u=( x1 , y 2 ),v=( x 2 , y 2 ) 由兩向量 u=( x1 , y 2 ),v=( x 2 , y 2 )其內積(點積)定義可求得:

若非零向量 u 和 v 是垂直(perpendicular)或正交的(orthogonal)若 且唯若內積 u.v=0 則知 cos  =0,即  = 90  ,稱 u,v 為正交向量 (a) X‧Y=0 cos  =0 →X,Y 正交 →X,Y 平行 (b) X  Y = X Y cos  =  1 (c) X  Y = X Y cos  =1 →X,Y 平行且同向

企二 D 管理數學講義 璽兒助教 2014/5/21

1

Chapter15.7~16.7
★ 內積(點積) Inner Product  a1   b1      n a 2  ,b= b2  a  b= a=  a 2 b2   a n bn   ai bi a1 b1    i 1     a n  bn      16.1 Determinant 定義和性質 ★A 為一n n之方陣,我們定義A的行列式為 det A | | ⋯

If A If A

a a a

, det A a a

|A| |A|

, det A

a a a a a a , det A |A| If A a a a ★ 行列式的特性 Properties of Determinants 設 A、B 為 n 階方陣 矩陣轉置行列式不改變 。 如果 A 藉由列互換後得到 B,則 。 如果 A 有兩列(行)相等,則 。 如果 A 有一列(行)全為零,則 。 如果 A 藉由乘上一實數 c 得到 B,則 。 如果 A 第 s 列(行)的每個元素乘以常數 c 再加到第 r 列(行) 所對應的元素,r≠s,而獲得 B,則 。 設 A 為一上(下)三角矩陣 upper(lower)triangular matrix 則 A 的行列式 ⋯ 以此類推:對角線矩陣的行列式=主對角線的元素相乘 將矩陣藉由基本列運算化為三角矩陣可簡化行列式的計算。 兩矩陣 AB 相乘的行列式=各自取行列式再相乘 。 設 A 為非奇異矩陣,則





基本列運算及基本行運算 第 i,j 兩列(行)互換: ↔ ↔ 第 i 列(行)乘以 K 0,以取代第 i 列(行):




2

企二 D 管理數學講義 璽兒助教 2014/5/21

Chapter15.7~16.7
第 i 列(行)乘以 K 0 加到第 j 列(行),以取代第 j 列(行): → → ★ 矩陣之基本列運算 如果 A 藉由列互換後得到 B


或 i j ↔ 如果 A 藉由乘上一實數 c(k)得到 B






K

0

如果 A 第 s 列(行)的每個元素乘以常數 c(k)再加到第 r 列(行)所對應的元素,r≠s,而獲得 B。


或 精選練習題: 1.若| | 2 B C 3 3 3 和 D 3



i

j

=3,求出下列矩陣的行列式值: 2 3 2 3

2. 將矩陣化簡成三角形矩陣以計算行列值 4 (a) 5 2 3 2 0 5 0 4

企二 D 管理數學講義 璽兒助教 2014/5/21

3

Chapter15.7~16.7
3.若 A 和 B 是n n階矩陣且|A| 2 和|B| 3,計算|A B |。

16.2 &16.5 Cofactor Expansion 餘因子展開式和應用
★ Minor 子行列式 A 為一 n  n 矩陣, A 的子行列式 後,取行列式的值。 ? , ? =刪除 A 矩陣 i 列 j 行

 2 1 3 EX A   1 2 0  , ○    3 2 1   



Cofactor 餘因子 A 為一方陣, A 的餘因子定義為 Aij  (1)i  j M ij EX 如上述矩陣 A,A ? , A ? ○



Computing Determinants using Cofactor 用餘因子展開計算行列式

 a11 a12 a13  det( A)   aij Aij ,設 A   aij  =  a21 a22 a23  ,則     j 1  a31 a32 a33    A  a11 A11  a12 A12  a13 A13  a11M11  a12 M12  a13 M13 n  1 2 3 EX A   4 2 1  ,用第一列展開: ○    2 0 2   
展開時需注意元素的正負號。 也可以使用 ERO(基本列運算)化簡後再計算
 例如對 A 進行 R121 後得到

 1 2 3 A   5 0 4  ,用第二行展開:    2 0 2   
企二 D 管理數學講義 璽兒助教 2014/5/21 4

Chapter15.7~16.7
 使用餘因子展開計算行列式時,用任一行或任一列都可以,一般是使 用基本列運算使某一行(列)得到最多的 0 再展開可以減少計算上的 時間,但要注意展開時元素的正負號,可使用符號表記憶:            2  2 可直接算不需展開, 3  3 時    , 4  4 時            以此類推。 A 為方陣。則對每個 1≦i≦n det A 又對每個 1≦j≦n det A ⋯ ⋯ (沿著第 i 列的 det(A)展開式): (沿著第 j 行的 det(A)展開式)



精選練習題:

1.使用定理計算行列式值: 4 4 2 1 1 2 0 3 2 0 3 4 0 3 2 1

16.7 the inverse of a matrix 逆矩陣
★ A、B 為 n 階方陣,若 AB=BA=In,B 為 A 的逆矩陣(inverse of A), 則 A 為非奇異矩陣(nonsingular matrix)或可逆矩陣(invertible matrix)。若 B 不存在,則稱 A 為奇異矩陣(Singular Matrix)或不可 逆矩陣(Noninvertible Matrix)。

★ 唯一性 (Uniqueness) 若方陣 A 可逆,則A 是唯一的,意即,可逆矩陣 A 僅有一個逆 矩陣。 若 A 的逆矩陣存在,則記作A 。因此 AA =A A=I ★ 逆矩陣的運算特性 (Properties of Inverse of Matrix)

-1 -1

=A




-1

= =B


-1



-1

AB

-1



-1

企二 D 管理數學講義 璽兒助教 2014/5/21

5

Chapter15.7~16.7
★ 求逆矩陣 將 A 寫成增廣矩陣的形式並加上一單位矩陣,如 A ⋮ I ,對 A 使 用基本列運算將之化為單位矩陣(左右兩個矩陣須進行同樣的運 算),若 A 可化為單位矩陣,此時會得到 I ⋮ B ,我們稱 B 為 A 的逆矩陣(可藉由兩矩陣相乘是否得到單位矩陣來驗證)。 ★ 逆矩陣與求解線性系統 求解線性系統 Ax  b 除了用高斯或高斯約旦消去法之外,也可以 先求係數矩陣 A 的反矩陣 A 1 ,再與常數矩陣 b 相乘得到線性系 統的解: x   1b ,此時得到的解是唯一解,若線性系統要有 唯一解,係數矩陣 A 必為一非奇異矩陣。(證明) ★ 齊次系統 Homogenous System 若有一線性系統其常數項皆為零,意即 x  0 ,我們稱此線性系 統為齊次系統,且存在有自然解(Trivial Solutions) x  0 。 若齊次系統要有自然解,係數矩陣 A 的反矩陣必須要存在(為一 非奇異矩陣)。 ★ 1. 2. 3. 4. Nonsingular Equivalence 以下敘述都是等價的 A 是非奇異矩陣 x  0 是 x  0 的唯一解 A 列等價於單位矩陣 I n 線性系統 Ax  b 只有唯一解

精選練習題: 1. 矩陣

1 3

1 4

,是奇異的或非奇異的?若它是非奇異的,求出它的逆矩

陣。

企二 D 管理數學講義 璽兒助教 2014/5/21

6

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...Math 1P05 Assignment #1 Due: September 26 Questions 3, 4, 6, 7, 11 and 12 require some Maple work. 1. Solve the following inequalities: a) b) c) 2. Appendix D #72 3. Consider the functions and . a) Use a Maple graph to estimate the largest value of at which the graphs intersect. Hand in a graph that clearly shows this intersection. b) Use Maple to help you find all solutions of the equation. 4. Consider the function. a) Find the domain of. b) Find and its domain. What is the range of? c) To check your result in b), plot and the line on the same set of axes. (Hint: To get a nice graph, choose a plotting range for bothand.) Be sure to label each curve. 5. Section 1.6 #62 6. Section 2.1 #4. In d), use Maple to plot the curve and the tangent line. Draw the secant lines by hand on your Maple graph. 7. Section 2.2 #24. Use Maple to plot the function. 8. Section 2.2 #36 9. Section 2.3 #14 10. Section 2.3 #26 11. Section 2.3 #34 12. Section 2.3 #36 Recommended Problems Appendix A all odd-numbered exercises 1-37, 47-55 Appendix B all odd-numbered exercises 21-35 Appendix D all odd-numbered exercises 23-33, 65-71 Section 1.5 #19, 21 Section 1.6 all odd-numbered exercises 15-25, 35-41, 51, 53 Section 2.1 #3, 5, 7 Section 2.2 all odd-numbered exercises 5-9, 15-25, 29-37 Section 2.3 all odd-numbered exercises...

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...find the national average cost of food for an individual, as well as for a family of 4 for a given month. http://www.cnpp.usda.gov/sites/default/files/usda_food_plans_cost_of_food/CostofFoodJan2012.pdf 5. Find a website for your local city government. http://www.usa.gov/Agencies/Local.shtml 6. Find the website for your favorite sports team (state what that team is as well by the link). http://blackhawks.nhl.com/ (Chicago Blackhawks) 7. Many of us do not realize how often we use math in our daily lives. Many of us believe that math is learned in classes, and often forgotten, as we do not practice it in the real world. Truth is, we actually use math every day, all of the time. Math is used everywhere, in each of our lives. Math does not always need to be thought of as rocket science. Math is such a large part of our lives, we do not even notice we are computing problems in our lives! For example, if one were interested in baking, one must understand that math is involved. One may ask, “How is math involved with cooking?” Fractions are needed to bake an item. A real world problem for baking could be as such: Heena is baking a cake that requires two and one-half cups of flour. Heena poured four and one-sixth cups of flour into a bowl. How much flour should Heena take out of the bowl? In this scenario of a real world problem, we have fractions, and subtraction of fractions, since Heena has added four and one-sixth cups of flour, rather than the needed...

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... h(x)= 7-x/3 First we need to compute (f-h)(4) (f*h)(4)=f(4)-h(4), each function can be done separately f(4)=2(4)+5 f(4)=8+5 f(4)=13 H h(4)=(7-4)/3 same process as above h(4)=3/3=h(4)=1 (f-h)(4)=13-1 (f-h)(4)=12 this is the solution after substituting and subtracting The next part we need to replace the x in the f function with the g (f*g)(x)=f(g(x)) (f*g)(x)=f(x2-3) (f*g)(x)=2x2-1 is the result Now we need to do the h function (h*g)(x)=h(g(x)) (h*g)(x)=h(x2-3) (h*g)(x)=7-(x2-3) (h*g)(x)=10-x2 end result The inverse function-- f-1(x)=x-5h-1(x)=-(3-7) By doing problems this way it can save a person and a business a lot of time. A lot of people think they don't need math everyday throughout their life, but in all reality people use math almost everyday in life. The more you know the better off your life will...

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