COMP 116: Introduction to Scientific Programming Lecture 7: Matrices and Linear Systems

Recap exercises Indexing: what does A(3:7,5:end) return? Creating: how do you create this matrix? Statistical functions for matrices: ◦ min, max, sum, mean, std

A gambling game One matrix M Two teams: the house and the gambler Gambler picks a column of M, house picks an element from this column. The gambler wins this amount ◦ How much should the house charge the gambler for playing this game? ◦ How much should the gambler pay for playing this game?

Today Linear systems ◦ Solving simultaneous (linear) equations

Simultaneous equations Solve this set of (math) equations ◦ 5x+7y=17 ◦ 2x+3y=7 Now solve this one ◦ 5x+7y+z+2w=30 ◦ 3x+4y+8z+w=39 ◦ 9x+y+4z+2w=31 ◦ 6x+2y+5z+8w=57

Linear Equations

Solving linear equations Ax=b ◦ A is an m x n matrix  Rows of A correspond to equations  Columns of A correspond to variables ◦ x is a n x 1 matrix (n element column vector) ◦ b is a m x 1 matrix ( m element column) In general if m equals n, then x=(A) -1 b ◦ Or in matlabspeak:  x=inv(A)*b  Matlab shorthand >> x=A\b

Exercise Solve these set of equations in the Matlab way ◦ Convert to Ax=b format ◦ Then x=inv(A)*b Set I ◦ 5x+7y=17 ◦ 2x+3y=7 Set II ◦ 5x+7y+z+2w=30 ◦ 3x+4y+8z+w=39 ◦ 9x+y+4z+2w=31 ◦ 6x+2y+5z+8w=57

Matrix multiplication

Vector multiplication >> [1 2 3]*[4;5;6] ans = 32 >> [1 2 3]*[4 5 6] ??? Error using ==> mtimes Inner matrix dimensions must agree. >> [1 2 3]*[4 5 6]' ans = 32

Matrix-Vector Multiplication

Linear Equations Key idea: Can write a complete linear system in vector notation: Ax=b. Since x is unknown, we need some way to express it in terms of A and b.

Matrix Math Matrix multiplication

Matrix Math Exercise Identity matrix

Matrix Math Inverse >> A*inv(A) ans = >> A\A ans = Inverse needs to exist.

Linear Equations: Solving One line: x = A\b Mathematical formulation MATLAB solution:

Matrix element naming convention a11a12a13a14 a21a22a23a24 a31a32a33a34