E. Non-homogeneous Linear Equations.
The system of non-homogeneous linear equations
has a solution if and only if the column vector
in the Space generated by the vectorsThis means that there is a solution if and only if the right column rank of the matrix is the same as the right column rank of the augmented matrix since the vector space generated by the original must be the same as the vector space generated by the augmented matrix and in either case the dimension is the same as the rank of the matrix by Theorem 2.
By Theorem 4, this means that the row tanks are equal. Conversely, if the row rank of the augmented matrix is the same as the row rank of the original matrix, the column ranks will be the same and the equations will have a solution.
If the equations (2) have a solution, then any relation among the rows of the original matrix subsists among the rows of the augmented matrix. For equations (2) this merely means that like combinations of equals are equal. Conversely, if each relation which subsists between the rows of the original matrix also subsists between the rows of the augmented matrix, then the row rank of the augmented matrix is the same as the row rank of the original matrix. In terms of the equations this means that there will exist a solution if and only if the equations are consistent, i.e., if and only if any dependence between the left hand sides of the equations also holds between the
THEOREM 5. If in equations (2) m = n, there exists a unique solution if and only if the corresponding homogeneous equations
atlXl+ a!2X2 + * * * + ainXn = 0
nl 1 «2 2 nil n
have only the trivial solution.
If they have only the trivial solution, then the column vectors are independent. It follows that the original n equations in n unknowns will have a unique solution if they have any solution, since the difference, term by term, of two distinct solutions would be a non-trivial solution of the homogeneous equations. A solution would exist since the n independent column vectors form a generating system for the n-dimensional space of column vectors.
Conversely, let us suppose our equations have one and only one solution. In this case, the homogeneous equations added term by term to a solution of the original equations would yield a new solution to the original equations. Hence, the homogeneous equations have only the trivial solution.