Question

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

Short answer

Yes, by using the rank theorem and the invertible matrix theorem.

Step-by-step solution

Describe the given statement

Consider the homogeneous system \(Ax = 0\), where A is the \(5 \times 6\) matrix. From the given statement, \({\rm{dim Null }}A = 1\).

Use the rank theorem

By the rank theorem, you get

\(\begin{aligned} {\rm{rank }}A &= n - {\rm{dim Null }}A\\ &= 6 - 1\\{\rm{rank }}A &= 5.\end{aligned}\)

As \({\rm{dim Col }}A = {\rm{rank }}A\), \({\rm{dim Col }}A = 5\). Since Col A is the subspace of \({\mathbb{R}^5}\), \({\rm{Col }}A = {\mathbb{R}^5}\).

Draw a conclusion

By the invertible matrix theorem, for every b in \({\mathbb{R}^5}\), the system \(Ax = b\) has a unique solution.