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Q26E

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Found in: Page 191

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

In statistical theory, a common requirement is that a matrix be of full rank. That is, the rank should be as large as possible. Explain why an m n matrix with more rows than columns has full rank if and only if its columns are linearly independent.

No, the new non-homogeneous system will not have any solution.

See the step by step solution

Step 1: Describe the given statement

It is given that a homogeneous system has full rank. It implies that if the system is $$m \times n$$, then the rank of the system is $$n$$ as the rank is equal to the number of pivot positions.

Step 2: Use the rank theorem

Consider the homogeneous system $$Ax = 0$$, where $$A$$ is an $$m \times n$$ matrix. As the system has full rank, $${\rm{rank }}A = n$$ and there are $$n$$ unknowns. By the rank theorem, $${\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n$$.

Put the values as shown:

\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= n - {\rm{rank}}\,A\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= n - n\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= 0\end{aligned}

Step 3: Draw a conclusion

As $${\rm{dim Nul}}\,A$$ is 0 and the system is homogeneous, there must exist only a trivial solution. This is possible when matrix $$A$$ has linearly independent columns.