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Chapter 1: Q11Q (page 1)
Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a line in \({\mathbb{R}^3}\).
The matrix that is not in the echelon form is \(\left( {\begin{aligned}{*{20}{c}}1&2&1\\1&5&2\end{aligned}} \right)\).
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Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).
Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).
Question: Determine whether the statements that follow are true or false, and justify your answer.
16: There exists a 2x2 matrix such that
.
Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.
18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)
Find the general solutions of the systems whose augmented matrices are given as
12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).
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