第 6 章線性代數方程式 Linear Algebraic Equations

第6章線性代數方程式Linear Algebraic Equations Puff! The magic dragon, live by the sea…

linear 5x-2y=13 7x+3y=24 • Linear ~ dependent variable 不可以有一次以外的次方 Puff! The magic dragon, live by the sea…

6.1 Elementary Solution Methods • Successive elimination of variables • Gauss elimination Puff! The magic dragon, live by the sea…

Gauss elimination -x+y+2z=2 (1) 3x-y+z=6 (2) -x+3y+4z=4 (3) -x+y+2z=2 (1) 2y+7z=12 (4) 2y+2z=2 (5) Puff! The magic dragon, live by the sea…

Gauss elimination -x+y+2z=2 (1) 5z=10 (6) y=-1 (7) x=1 Puff! The magic dragon, live by the sea…

Self test p.334 Puff! The magic dragon, live by the sea…

Singular & ill-conditioned problems • 3x-4y=5 6x-10y=2 Sol. (7,4) • 3x-4y=5 6x-8y=10 Infinite solutions • 3x-4y=5 6x-8y=3 empty set Puff! The magic dragon, live by the sea…

Puff! The magic dragon, live by the sea…

Homogeneous equations 6x+ay=0 2x+4y=0  (a-12)y=0  If a~=12, x=0 If a=12, infinite solutions Puff! The magic dragon, live by the sea…

Ill-conditioned equations 3x-4y=5 (1) 6x-8.002y=3 (2) •  y=(3x-5)/4 •  y=(3x-1.5)/4.001 (x,y)=(4668, 3500) Very sensitive ! Puff! The magic dragon, live by the sea…

Matrix methods for linear equations Puff! The magic dragon, live by the sea…

Determinants 行列式 A=[3,-4,1;6,10,2;9,-7,3]; det(A) ans = 0 Puff! The magic dragon, live by the sea…

Determinants and singular problems • If determinant of A=0, then a singular problem. • If determinant of A ~= 0 • If determinant of A = 0 Puff! The magic dragon, live by the sea…

Left-division method A=[3,2,-9;-9,-5,2;6,7,3]; det(A) b=[-65;16;5]; A\b ans = 288 ans = 2.0000 -4.0000 7.0000 Puff! The magic dragon, live by the sea…

Right-division method Puff! The magic dragon, live by the sea…

Ex.6.2-2 p.342 • Kirchhoff’s law • Current conservation at a node Puff! The magic dragon, live by the sea…

% resist.m % solve the current i_1,i_2,i_3 R=[5,100,200,150,250]*1000; v1=100; v2=50; A1=[R(1)+R(4),-R(4),0]; A2=[-R(4),R(2)+R(4)+R(5),-R(5)]; A3=[0,R(5),-(R(3)+R(5))]; A=[A1;A2;A3]; b=[v1;0;v2]; current=A\b; disp('The currents are:') disp(current) The currents are: 1.0e-003 * 0.9544 0.3195 0.0664 Puff! The magic dragon, live by the sea…

Exercise p.343-345 Puff! The magic dragon, live by the sea…

Matrix inverse Puff! The magic dragon, live by the sea…

An example of Statics • Derivation of equilibrium equations. • Solve the linear equations by matlab. • P.346-348 Puff! The magic dragon, live by the sea…

Ex. 6.2-5 • Check by matlab as well as by hand for an inversion of 2 by 2 matrix. 2x+9y=5 3x-4y=7 A=[2,9;3,-4]; b=[5;7]; x=inv(A)*b x = 2.3714 0.0286 Puff! The magic dragon, live by the sea…

Cramer’s method • Self test … Puff! The magic dragon, live by the sea…

6.4 Underdetermined systems A=[1,3]; b=6; x=A\b x = 0 2 Puff! The magic dragon, live by the sea…

Underdetermined systems Puff! The magic dragon, live by the sea…

Matrix rank 矩陣的秩 A=[3,-4,1;6,10,2;9,-7,3]; rank(A) ans = 2 Puff! The magic dragon, live by the sea…

Existence and uniqueness of solutions • Augmented matrix Puff! The magic dragon, live by the sea…

Singular matrix Puff! The magic dragon, live by the sea…

Ex 6.4-1 3x-2y+8z=48 -6x+5y+z=-12 9x+4y+2z=24 A=[3,-2,8;-6,5,1;9,4,2]; b=[48;-12;24]; rank(A) rank([A b]) x=A\b ans = 3 ans = 3 x = 2.0000 -1.0000 5.0000 Puff! The magic dragon, live by the sea…

Ex 6.4-2 2x-4y+5z=-4 -4x-2y+3z=4 2x+6y-8z=0 • A=[2,-4,5;-4,-2,3;2,6,-8]; • b=[-4;4;0]; • rank(A) • rank([A b]) • x=A\b • ans = • 2 • ans = • 2 • Warning: Matrix is singular to working precision. • x = • Inf • Inf • Inf Puff! The magic dragon, live by the sea…

pinv command & Euclidean norm • x=pinv(A)*b x = -1.2148 0.2074 -0.1481 Puff! The magic dragon, live by the sea…

Ex 6.4-3 • Statically indeterminate problem Puff! The magic dragon, live by the sea…

pinv(A)*b ans = 93.3333 133.3333 173.3333 A=[1,1,1;10,5,0]; b=[400;1600]; rank(A) rank([A b]) A\b ans = 2 ans = 2 ans = 160.0000 0 240.0000 Puff! The magic dragon, live by the sea…

Ex 6.4-4, pp.362 A=[2,-4,5;-4,-2,3;2,6,-8]; b=[-4;4;0]; x=pinv(A)*b rref([A b]) x = -1.2148 0.2074 -0.1481 ans = 1.0000 0 -0.1000 -1.2000 0 1.0000 -1.3000 0.4000 0 0 0 0 2x-4y+5z=-4 -4x-2y+3z=4 2x+6y-8z=0 • x+0y-0.1z=-1.2 • 0x+y-1.3z=0.4 • 0x+0y+0z=0 Puff! The magic dragon, live by the sea…

Underdetermined systems • P.363-364 ex.6.4-5 5x+3y+3z=40 3x+3y+4z=30 • x-0.5z=5 • y+1.8333z=5 A=[5,3,3;3,3,4]; b=[40;30]; rref([A b]) ans = 1.0000 0 -0.5000 5.0000 0 1.0000 1.8333 5.0000 Puff! The magic dragon, live by the sea…

Ex 6.4-6 p.364 100+200=f1+f4 f1+f2=300+200 600+400=f2+f3 f3+f4=300+500 Puff! The magic dragon, live by the sea…

第 6 章線性代數方程式 Linear Algebraic Equations