Equations in two unknowns

This vignette illustrates the ideas behind solving systems of linear equations of the form \(\mathbf{A x = b}\) where

\(\mathbf{A}\) is an \(m \times n\) matrix of coefficients for \(m\) equations in \(n\) unknowns
\(\mathbf{x}\) is an \(n \times 1\) vector unknowns, \(x_1, x_2, \dots x_n\)
\(\mathbf{b}\) is an \(m \times 1\) vector of constants, the “right-hand sides” of the equations

The general conditions for solutions are:

the equations are consistent (solutions exist) if \(r( \mathbf{A} | \mathbf{b}) = r( \mathbf{A})\)
- the solution is unique if \(r( \mathbf{A} | \mathbf{b}) = r( \mathbf{A}) = n\)
- the solution is underdetermined if \(r( \mathbf{A} | \mathbf{b}) = r( \mathbf{A}) < n\)
the equations are inconsistent (no solutions) if \(r( \mathbf{A} | \mathbf{b}) = r( \mathbf{A})\)

We use c( R(A), R(cbind(A,b)) ) to show the ranks, and all.equal( R(A), R(cbind(A,b)) ) to test for consistency.

library(matlib)   # use the package

Equations in two unknowns

Each equation in two unknowns corresponds to a line in 2D space. The equations have a unique solution if all lines intersect in a point.

Two consistent equations

A <- matrix(c(1, 2, -1, 2), 2, 2)
b <- c(2,1)
showEqn(A, b)

##      [,1]               
## [1,] "1*x1 - 1*x2  =  2"
## [2,] "2*x1 + 2*x2  =  1"

c( R(A), R(cbind(A,b)) )          # show ranks

## [1] 2 2

all.equal( R(A), R(cbind(A,b)) )  # consistent?

## [1] TRUE

Plot them:

plotEqn(A,b)

plot of chunk unnamed-chunk-3

Three consistent equations

For three (or more) equations in two unknowns, \(r(\mathbf{A}) \le 2\), because \(r(\mathbf{A}) \le \min(m,n)\). The equations will be consistent if \(r(\mathbf{A}) = r(\mathbf{A | b})\), which means that whatever linear relations exist among the rows of \(\mathbf{A}\) are the same as those among the elements of \(\mathbf{b}\).

A <- matrix(c(1,2,3, -1, 2, 1), 3, 2)
b <- c(2,1,3)
showEqn(A, b)

##      [,1]               
## [1,] "1*x1 - 1*x2  =  2"
## [2,] "2*x1 + 2*x2  =  1"
## [3,] "3*x1 + 1*x2  =  3"

c( R(A), R(cbind(A,b)) )          # show ranks

## [1] 2 2

all.equal( R(A), R(cbind(A,b)) )  # consistent?

## [1] TRUE

Plot them:

plotEqn(A,b)

plot of chunk unnamed-chunk-5

Three inconsistent equations

A <- matrix(c(1,2,3, -1, 2, 1), 3, 2)
b <- c(2,1,6)
showEqn(A, b)

##      [,1]               
## [1,] "1*x1 - 1*x2  =  2"
## [2,] "2*x1 + 2*x2  =  1"
## [3,] "3*x1 + 1*x2  =  6"

c( R(A), R(cbind(A,b)) )          # show ranks

## [1] 2 3

all.equal( R(A), R(cbind(A,b)) )  # consistent?

## [1] "Mean relative difference: 0.5"

An approximate solution is sometimes available using a generalized inverse.

x <- MASS::ginv(A) %*% b
x

##      [,1]
## [1,]    2
## [2,]   -1

Plot the equations. You can see that each pair of equations has a solution, but all three do not have a common, consistent solution.

plotEqn(A,b, xlim=c(-2, 4))
points(x[1], x[2], pch=15)

plot of chunk unnamed-chunk-8

Equations in three unknowns

Each equation in three unknowns corresponds to a plane in 3D space. The equations have a unique solution if all planes intersect in a point.

Three consistent equations

A <- matrix(c(2, 1, -1,
             -3, -1, 2,
             -2,  1, 2), 3, 3, byrow=TRUE)
colnames(A) <- paste0('x', 1:3)
b <- c(8, -11, -3)
showEqn(A, b)

##      [,1]                         
## [1,] "2*x1 + 1*x2 - 1*x3  =  8"   
## [2,] "-3*x1 - 1*x2 + 2*x3  =  -11"
## [3,] "-2*x1 + 1*x2 + 2*x3  =  -3"

Are the equations consistent?

c( R(A), R(cbind(A,b)) )          # show ranks

## [1] 3 3

all.equal( R(A), R(cbind(A,b)) )  # consistent?

## [1] TRUE

Solve for \(\mathbf{x}\).

solve(A, b)

## x1 x2 x3 
##  2  3 -1

solve(A) %*% b

##    [,1]
## x1    2
## x2    3
## x3   -1

inv(A) %*% b

##      [,1]
## [1,]    2
## [2,]    3
## [3,]   -1

Another way to see the solution is to reduce \(\mathbf{A | b}\) to echelon form. The result is \(\mathbf{I | A^{-1}b}\), with the solution in the last column.

echelon(A, b)

##      x1 x2 x3   
## [1,]  1  0  0  2
## [2,]  0  1  0  3
## [3,]  0  0  1 -1

echelon(A, b, verbose=TRUE, fractions=TRUE)

## row: 1 
##      x1   x2   x3       
## [1,]    1  1/3 -2/3 11/3
## [2,]    0  1/3  1/3  2/3
## [3,]    0  5/3  2/3 13/3
## row: 2 
##      x1   x2   x3       
## [1,]    1    0 -4/5 14/5
## [2,]    0    1  2/5 13/5
## [3,]    0    0  1/5 -1/5
## row: 3 
##      x1 x2 x3   
## [1,]  1  0  0  2
## [2,]  0  1  0  3
## [3,]  0  0  1 -1

##      x1 x2 x3   
## [1,]  1  0  0  2
## [2,]  0  1  0  3
## [3,]  0  0  1 -1

Plot them. plotEqn3d uses rgl for 3D graphics. If you rotate the figure, you'll see an orientation where all three planes intersect at the solution point, \(\mathbf{x} = (2, 3, -1)\)

plotEqn3d(A,b, xlim=c(0,4), ylim=c(0,4))

plotEqn1snapshot
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Three inconsistent equations

A <- matrix(c(1,  3, 1,
              1, -2, -2,
              2,  1, -1), 3, 3, byrow=TRUE)
colnames(A) <- paste0('x', 1:3)
b <- c(2, 3, 6)
showEqn(A, b)

##      [,1]                      
## [1,] "1*x1 + 3*x2 + 1*x3  =  2"
## [2,] "1*x1 - 2*x2 - 2*x3  =  3"
## [3,] "2*x1 + 1*x2 - 1*x3  =  6"

Are the equations consistent? No.

c( R(A), R(cbind(A,b)) )          # show ranks

## [1] 2 3

all.equal( R(A), R(cbind(A,b)) )  # consistent?

## [1] "Mean relative difference: 0.5"