Let A be an n × n matrix with rank r(0 < r < n). Then Ax = 0 has p independent solutions, where p is
Let A be an n × n matrix with rank r(0 < r < n). Then Ax = 0 has p independent solutions, where p is Correct Answer n - r
Concept:
Homogeneous system AX = 0
Non – homogeneous solution AX_{{\rm{m}} \times {\rm{n}}}}{\left_{{\rm{n}} \times 1}} = {\left_{{\rm{m}} \times 1}}\)
‘m’ = number of equations
‘n'= number of unknowns
Trivial solution (or) zero solution (X) = 0
Non – trivial solution (X) [ height="20" src="file:///C:/Users/abhayraj/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png" width="11"> 0
Note:
1. Every homogeneous equation is always consistent with the trivial solution.
2. A non – trivial solution may or may not exist. But if exists infinite number only.
Linear dependence: Two vectors X, Y are said to be linearly dependent if X = α Y or Y=α X. Otherwise they are linearly independent.
Null-space:
The nullspace of the matrix A is denoted by N(A). It is the set of all n-dimensional column vectors x such that Ax = 0.
Nullity is defined as the dimension of null space
= number of linearly independent solutions
Nullity = no.of unknowns – the rank of A = n – r
