Question

Suppose the solutions of a homogeneous system of five linear equationsin six unknowns are all multiples of one nonzero solution. Will thesystem necessarily have a solution for every possible choice of constantson the right sides of the equations? Explain.

Answer 1

Explanation:

The problem can be written as

Ar = where A = 5×6 matrix. dim(NullA)=1 as all the multiples of Ar=0

are the multiple of one non zero solution.

Since A is 5×6 , A n= 6

rank(A)= n-dim(Nul A)= n-1= 6-1=5

now, imagine A as 5 dimensional subspace of R^5( As A has 5 rows)

Therefore Col(A)= R^5

This means that Ar = b has a solution for every b.