Whether it is possible to find two non-zero solutions of the associated homogenous m use system that are not the multiple of each other.
The dimension of the column space and the row space of an mxn matrix A are equal. This common dimension, the rank of A, also equals to the number of pivot positions in A and satisfies the equation Rank(A)+dim Nul(A)=n.
Explanation: It is given that a non-homogeneous system of 9 linear equations and 10 unknowns has a solution for all possible constants on the right hand side of the equation.
Consider the system of equation Ax=z
The matrix A of order m*nrepresents it contain m rows and n columns.
Here, A is a \(9 xx 10\) matrix.
Here, RankA =9 as the system of equation has solution for all \(z in R^9\).
Rank(A)+ dim Nul(A) =n dim(nul4)+RankA =10
dim(nul4) = 10-9
dim(nulA) = 1
That is, it is impossible to find two linearly independent vectors in NullA.