Linear Equation System Engineering Mathematics I

Linear Equation System Engineering Mathematics I Augmented matrix A

Upper triangular matrix Gauss Elimination (1) Eliminate Engineering Mathematics I Upper triangular matrix

Gauss Elimination (2) Backward substitution 4 Engineering Mathematics I Backward substitution 4

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

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

Possibilities (1) Linear equation system has three possibilities of solutions Engineering Mathematics I

Engineering Mathematics I

Example 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

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

Linear Independence Let a1, …, am be any vectors in a vector space V. Then an expression of the form c1a1 + … + cmam (c1, …, cm any scalars) is called linear combination of these vectors. The set S of all these linear combinations is called the span of a1, …, am. Consider the equation: c1a1 + … + cmam = 0 If the only set of scalars that satisfies the equation is c1 = … = cm = 0, then the set of vectors a1, …, am are linearly independent. Engineering Mathematics I

Linear Dependence Otherwise, if the equation also holds with scalars c1, …, cm 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 c1 ≠ 0, a1 = l2a2 + … + lmam where lj = -cj/c1 Engineering Mathematics I

Example 3 Consider the vectors: i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1], and the equation: c1i + c2j + c3k = 0 Then: [(c1i1+c2j1+c3k1), (c1i2+c2j2+c3k2), (c1i3+c2j3+c3k3)] = 0 [c1i1, c2j2, c3k3] = 0 c1 = c2 = c3 = 0 Consider vectors a = [1, 2, 1], b = [0, 0, 3], d = [2, 4, 0]. Are they linearly independent? Engineering Mathematics I

Rank of a Matrix 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

Example 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

Example 5 Engineering Mathematics I

Example 6 Engineering Mathematics I

Example 7 Engineering Mathematics I

Some Notes For a single vector a, then the equation ca = 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

Rank of a Matrix (2) The rank of a matrix A equals the maximum number of linearly independent column vectors of A. Hence A and AT 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 R3. R3 has 3 dimension and i, j, k is the basis of R3. Engineering Mathematics I

General Properties of Solutions 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