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Natural Equations
Linear Functions
Arrays
Needed by:
Real Linear Equation Solutions
Real Matrices
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Real Linear Equations

Why

Linear equations are ubiquitous.

Definition

Given aRn and yR, suppose we want to find xRn satisfying

a1x1+a2x2++anxn=y.

We refer to this expression as a real linear equation or linear equation. We treat each component xiR as a variable and we treat each component aiR and yR as constants. We call the pair (a,y) the real linear equation constants.1

The source of the terminology “linear” is by viewing the left hand side as a function. Define f:RnR by f(x)=iaixi. We want to find xRn to satisfy f(x)=b. Notice that f is a linear real function.2

Moreover, to each linear function f:RdR there exists a vector aRd so that f(x)=iaixi. For this reason, if we are given several linear function f1,,fm, then we can think of these as several vectors a1,,an. If we are also given biR for each i=1,,m, then we have the vector bRm

We can define the two-dimensional array ARm×n by Aij=aji. For this reason, a linear system of equations is a pair (A,b). A solution of a linear system of equations is a vector xRn satisfying the equations

A11x1+A12x2++A1nxn=b1A21x1+A22x2++A2nxn=b2Am1x1+Am2x2++Amnxn=bn

Other terminology includes a system of linear equations or linear system or simultaneous linear equations


  1. Future editions will clarify. ↩︎
  2. Future editions may require a sheet here. ↩︎
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