Morpho (version 2.1)

solutionSpace: returns the solution space (basis and translation vector) for an equation system

Description

returns the solution space (basis and translation vector) for an equation system

Usage

solutionSpace(A, b)

Arguments

A

numeric matrix

b

numeric vector

Value

• basis
  matrix containing the basis of the solution space
• translate
  translation vector

Details

For a linear equationsystem, $Ax = b$, the solution space then is $$x = A^* b + (I - A^* A) y$$ where $A^*$ is the Moore-Penrose pseudoinverse of $A$. The QR decomposition of $I - A^* A$ determines the dimension of and basis of the solution space.

Examples

A <- matrix(rnorm(21),3,7)
b <- c(1,2,3)
subspace <- solutionSpace(A,b)
dims <- ncol(subspace$basis) # we now have a 4D solution space
## now pick any vector from this space. E.g
y <- 1:dims
solution <- subspace$basis%*%y+subspace$translate # this is one solution for the equation above
A%*%solution ## pretty close