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L INEAR E QUATION S YSTEM Engineering Mathematics I

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

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

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

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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

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

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

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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

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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

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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

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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

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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

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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

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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

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E XAMPLE 5 Engineering Mathematics I 16

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E XAMPLE 6 Engineering Mathematics I 17

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E XAMPLE 7 Engineering Mathematics I 18

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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

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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

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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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