Lecture 3 Linear Equations and Matrices - Stanford Engineering Everywhere
Lecture 3 Linear Equations and Matrices
? linear functions ? linear equations ? solving linear equations
3?1
Linear functions
function f maps n-vectors into m-vectors is linear if it satisfies:
? scaling : for any n-vector x, any scalar , f (x) = f (x)
? superposition: for any n-vectors u and v, f (u + v) = f (u) + f (v)
x1 example: f (x) = y, where x = x2 , y =
x3
let's check scaling property:
x3 - 2x1 3x1 - 2x2
f (x) =
(x3) - 2(x1) 3(x1) - 2(x2)
=
x3 - 2x1 3x1 - 2x2
= f (x)
Linear Equations and Matrices
3?2
Matrix multiplication and linear functions
general example: f (x) = Ax, where A is m ? n matrix ? scaling: f (x) = A(x) = Ax = f (x) ? superposition: f (u + v) = A(u + v) = Au + Av = f (u) + f (v) so, matrix multiplication is a linear function
converse: every linear function y = f (x), with y an m-vector and x and n-vector, can be expressed as y = Ax for some m ? n matrix A you can get the coefficients of A from Aij = yi when x = ej
Linear Equations and Matrices
3?3
Composition of linear functions
suppose
? m-vector y is a linear function of n-vector x, i.e., y = Ax where A is m?n
? p-vector z is a linear function of y, i.e., z = By where B is p ? m.
then z is a linear function of x, and z = By = (BA)x so matrix multiplication corresponds to composition of linear functions, i.e., linear functions of linear functions of some variables
Linear Equations and Matrices
3?4
Linear equations
an equation in the variables x1, . . . , xn is called linear if each side consists of a sum of multiples of xi, and a constant, e.g.,
1 + x2 = x3 - 2x1
is a linear equation in x1, x2, x3
any set of m linear equations in the variables x1, . . . , xn can be represented by the compact matrix equation
Ax = b,
where A is an m ? n matrix and b is an m-vector
Linear Equations and Matrices
3?5
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