Consider the provided question,

Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.

It can be written as,

Ax=0 where A=5*6 matrix

dim(NulA)=1 because all solutions of Ax=0 are multiples of nonzero solution

We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.

Since,A is 5*6 matrix n=6

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

Image of A is 5 dimensional subspace of \(R^5\) (because A has 5 rows)

So,Col(A)=\(R^5\)

This means that Ax=b has a solution for every b

Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.

It can be written as,

Ax=0 where A=5*6 matrix

dim(NulA)=1 because all solutions of Ax=0 are multiples of nonzero solution

We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.

Since,A is 5*6 matrix n=6

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

Image of A is 5 dimensional subspace of \(R^5\) (because A has 5 rows)

So,Col(A)=\(R^5\)

This means that Ax=b has a solution for every b