Suggested languages for you:

Americas

Europe

Q29E

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

**Use Exercise 28 to explain why the equation **\(Ax = b\)**has a solution for all **\({\rm{b}}\)** in **\({\mathbb{R}^m}\)** if and only if the equation **\({A^T}x = 0\)**has only the trivial solution.**

For the condition given, the number of non-pivot columns in \({A^T}\) is 0, which implies that \({A^T}x = 0\) has a trivial solution.

Consider the nonhomogeneous system \(Ax = b\) with matrix \(A\) of the size \(m \times n\). If the system has a pivot position in each row, it will have solution for all values of \(b\) in the subspace of \({\mathbb{R}^m}\) .

If the system \(Ax = b\) has a pivot position in each row, then the number of columns in \(A\) is \(m\), that is, \({\rm{dimCol}}\,\,A = m\). By the result of Exercise 28b,\({\rm{dimCol}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = m\). Put the values in \({\rm{dimCol}}\,\,A = m\) to get

\(\begin{aligned} {\rm{dim}}\,{\rm{Col}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= m\\m + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= m\\{\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= 0.\end{aligned}\)

As \({\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = 0\), the number of non-pivot columns in \({A^T}\) is 0, which implies that \({A^T}x = 0\) has a trivial solution.

94% of StudySmarter users get better grades.

Sign up for free