# Solving Linear Equations" In matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

knitr::opts_chunk$set( warning = FALSE, message = FALSE ) options(digits=4) par(mar=c(5,4,1,1)+.1)  library(rgl) library(knitr) knit_hooks$set(webgl = hook_webgl)


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


Plot the equations:

par(mar=c(4,4,0,0)+.1)
plotEqn(A,b)


Solve() is a convenience function that shows the solution in a more comprehensible form:

Solve(A, b, fractions = TRUE)


### 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 $r(\mathbf{A}) = r(\mathbf{A | b})$. This means that whatever linear relations exist among the rows of $\mathbf{A}$ are the same as those among the elements of $\mathbf{b}$.

Geometrically, this means that all three lines intersect in a point.

A <- matrix(c(1,2,3, -1, 2, 1), 3, 2)
b <- c(2,1,3)
showEqn(A, b)
c( R(A), R(cbind(A,b)) )          # show ranks
all.equal( R(A), R(cbind(A,b)) )  # consistent?

Solve(A, b, fractions=TRUE)       # show solution


Plot the equations:

par(mar=c(4,4,0,0)+.1)
plotEqn(A,b)


### Three inconsistent equations

Three equations in two unknowns are inconsistent when $r(\mathbf{A}) < r(\mathbf{A | b})$.

A <- matrix(c(1,2,3, -1, 2, 1), 3, 2)
b <- c(2,1,6)
showEqn(A, b)
c( R(A), R(cbind(A,b)) )          # show ranks
all.equal( R(A), R(cbind(A,b)) )  # consistent?


You can see this in the result of reducing $\mathbf{A} | \mathbf{b}$ to echelon form, where the last row indicates the inconsistency.

echelon(A, b)


Solve() shows this more explicitly:

Solve(A, b, fractions=TRUE)


An approximate solution is sometimes available using a generalized inverse.

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


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

par(mar=c(4,4,0,0)+.1)
plotEqn(A,b, xlim=c(-2, 4))
points(x[1], x[2], pch=15)


## 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)


Are the equations consistent?

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


Solve for $\mathbf{x}$.

solve(A, b)
solve(A) %*% b
inv(A) %*% b


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


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))


### 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)


Are the equations consistent? No.

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


## Try the matlib package in your browser

Any scripts or data that you put into this service are public.

matlib documentation built on April 4, 2018, 5:03 p.m.