self doubt

Question

For the system of linear equation Ax=0 where matrix A(mxn) , what can we say about the number of solutions for this equation :

1. if all n columns of A are linearly Independent.

2. if less than n columns of A are linearly Independent.

Answer 1

1. If all n columns of A are linearly Independent

So only 2 cases are possible for m&n i.e m<n or m=n 

when m=n,

Columns of A will always have an unique solution. (Columns of A fills the Space of R^m)

when m>n,

Columns of A will either have an unique solution or no solution (in case of Ax=b)

but since here we are dealing with Ax=0, we always have a unique solution which is the Trivial Solution (A.0 = 0)

2. if less than n columns of A are Linearly Independent

This is means the set of columns of A is linearly dependent.

Therefore, columns of A will always have more than one solution (infinite solutions).

 

Comments

@SumitWizardDas m<n condition is not possible when n LI columns are there

Answer 2

1. if all n columns of A are linearly Independent.
then rank of A is n and nuility is 0, The only solution to 𝐴𝑥=0 is the trivial solution x=0. This means that the system has a unique solution (the trivial one)
2. if less than n columns of A are linearly Independent.
then nullity(𝐴)=𝑛−rank(𝐴)>0, he system 𝐴𝑥=0Ax=0 has infinitely many solutions.
refer the following tree diagram for better and easy understanding.