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L INEAR E QUATION S YSTEM Engineering Mathematics I.

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Presentation on theme: "L INEAR E QUATION S YSTEM Engineering Mathematics I."— Presentation transcript:

1 L INEAR E QUATION S YSTEM Engineering Mathematics I

2 L INEAR E QUATION S YSTEM Engineering Mathematics I 2 Augmented matrix A

3 G AUSS E LIMINATION (1) Eliminate Engineering Mathematics I 3 Upper triangular matrix

4 G AUSS E LIMINATION (2) Engineering Mathematics I Backward substitution 4 4

5 E XAMPLE 1 Pivot element * Replace 2 nd eq.  (2 nd eq.) – 2x(1 st eq.) * Replace 3 rd eq.  (3 rd eq.) + 1x(1 st eq.) Engineering Mathematics I 5

6 E XAMPLE 1 Engineering Mathematics I 6 * Replace 3 rd eq.  (3 rd eq.) + 3x(2 nd eq.)  Upper triangle

7 P OSSIBILITIES (1) Linear equation system has three possibilities of solutions Engineering Mathematics I 7

8 8

9 E XAMPLE 2 Kirchhoff's current Law (KCL): At any point of a circuit, the sum of the inflowing currents equals the sum of out flowing currents. Kirchhoff's voltage law (KVL): In any closed loop, the sum of all voltage drops equals the impressed electromotive force. Engineering Mathematics I 9

10 E XAMPLE #2 Node P: i1 – i2 + i3 = 0 Node Q: -i1 + i2 –i3 = 0 Right loop: 10i2 + 25i3 = 90 Left loop: 20i1 + 10i2 = 80 Engineering Mathematics I 10

11 L INEAR I NDEPENDENCE Let a 1, …, a m be any vectors in a vector space V. Then an expression of the form c 1 a 1 + … + c m a m (c 1, …, c m any scalars) is called linear combination of these vectors. The set S of all these linear combinations is called the span of a 1, …, a m. Consider the equation: c 1 a 1 + … + c m a m = 0 If the only set of scalars that satisfies the equation is c 1 = … = c m = 0, then the set of vectors a 1, …, a m are linearly independent. Engineering Mathematics I 11

12 L INEAR D EPENDENCE Otherwise, if the equation also holds with scalars c 1, …, c m not all zero (at least one of them is not zero), we call this set of vectors linearly dependent. Linear dependent  at least one of the vectors can be expressed as a linear combination of the others. If c 1 ≠ 0, a 1 = l 2 a 2 + … + l m a m where l j = -c j /c 1 Engineering Mathematics I 12

13 E XAMPLE 3 Consider the vectors: i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1], and the equation: c 1 i + c 2 j + c 3 k = 0 Then: [(c 1 i 1 +c 2 j 1 +c 3 k 1 ), (c 1 i 2 +c 2 j 2 +c 3 k 2 ), (c 1 i 3 +c 2 j 3 +c 3 k 3 )] = 0 [c 1 i 1, c 2 j 2, c 3 k 3 ] = 0 c 1 = c 2 = c 3 = 0 Consider vectors a = [1, 2, 1], b = [0, 0, 3], d = [2, 4, 0]. Are they linearly independent? Engineering Mathematics I 13

14 R ANK OF A M ATRIX There are some possibilities of solutions of linear equation system: no solution, single solution, many solution. Rank of matrix  a tool to observe the problems of existence and uniqueness. The maximum number of linearly independent row vectors of a matrix A is called the rank of A. Rank A = 0, if and only if A = 0. Engineering Mathematics I 14

15 E XAMPLE 4 Matrix A above has rank A = 2 Since the last row is a linear combination of the two others (six times the first row minus ½ times the second), which are linearly independent. Engineering Mathematics I 15

16 E XAMPLE 5 Engineering Mathematics I 16

17 E XAMPLE 6 Engineering Mathematics I 17

18 E XAMPLE 7 Engineering Mathematics I 18

19 S OME N OTES For a single vector a, then the equation c a = 0, is satisfied if: c = 0, and a ≠ 0  a is linearly independent a = 0, there will be some values c ≠ 0  a is linearly dependent. Rank A = 0, if and only if A = 0. Rank A = 0  maximum number of linearly independent vectors is 0. If A = 0, there will be some values c1, …, cm which are not equal to 0, then vectors in A are linearly dependent. Engineering Mathematics I 19

20 R ANK OF A M ATRIX (2) The rank of a matrix A equals the maximum number of linearly independent column vectors of A. Hence A and A T have the same rank. If a vector space V is such that it contains a linearly independent set B of n vectors, whereas any set of n + 1 or more vectors in V is linearly dependent, then V has n dimension and B is called a basis of V. Previous example: vectors i, j, and k in vector space R 3. R 3 has 3 dimension and i, j, k is the basis of R 3. Engineering Mathematics I 20

21 G ENERAL P ROPERTIES OF S OLUTIONS A system of m linear equations has solutions if and only if the coefficient matrix A and the augmented matrix Ã, have the same rank. If this rank r equals n, the system has one solution. If r < n, the system has infinitely many solutions. Engineering Mathematics I 21


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