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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

Expert verified

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

Step by step solution

01

Describe the given statement

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

02

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 \({\mathbb{R}^5}\), \({\rm{Col }}A = {\mathbb{R}^5}\).

03

Draw a conclusion

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

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