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Q26E

Expert-verifiedFound in: Page 191

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.** **

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.

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:

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

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.

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