Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 29E from Chapter 1.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 29E
Chapter:
Problem:
Construct a 3 × 3 matrix, not in echelon form, whose columns span ℝ3. Show that the matrix you construct has the desired property.
Step-by-Step Solution
Given Information
We have to construct a $3\times3$ matrix that is not in echelon form, whose columns span $R^3$
Step-1:
Let the matrix be: \[A = \left[ {\begin{array}{*{20}{l}} 0&0&2\\ 0&3&0\\ 4&0&0 \end{array}} \right]\] For the above matrix, each Leading entry of a row is not in a column to the right of the leading entry of the row above it. Therefore, the matrix is not in echelon form.
Step-2:
Let there be an arbitrary vector in $R^3$ \[{\bf{v}} = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right]\] The given vector can be written as a linear combination of columns of the chosen matrix as shown below: \[{\bf{v}} = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right] = \dfrac{c}{2}\left[ {\begin{array}{*{20}{l}} 0\\ 0\\ 2 \end{array}} \right] + \dfrac{b}{2}\left[ {\begin{array}{*{20}{l}} 0\\ 2\\ 0 \end{array}} \right] + \dfrac{a}{2}\left[ {\begin{array}{*{20}{l}} 2\\ 0\\ 0 \end{array}} \right]\] Therefore, the columns of chosen matrix span $R^3$
\[\left[ {\begin{array}{*{20}{l}} 0&0&2\\ 0&2&0\\ 2&0&0 \end{array}} \right]\]
We have to construct a $3\times3$ matrix that is not in echelon form, whose columns span $R^3$
Step-1:
Let the matrix be: \[A = \left[ {\begin{array}{*{20}{l}} 0&0&2\\ 0&3&0\\ 4&0&0 \end{array}} \right]\] For the above matrix, each Leading entry of a row is not in a column to the right of the leading entry of the row above it. Therefore, the matrix is not in echelon form.
Step-2:
Let there be an arbitrary vector in $R^3$ \[{\bf{v}} = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right]\] The given vector can be written as a linear combination of columns of the chosen matrix as shown below: \[{\bf{v}} = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right] = \dfrac{c}{2}\left[ {\begin{array}{*{20}{l}} 0\\ 0\\ 2 \end{array}} \right] + \dfrac{b}{2}\left[ {\begin{array}{*{20}{l}} 0\\ 2\\ 0 \end{array}} \right] + \dfrac{a}{2}\left[ {\begin{array}{*{20}{l}} 2\\ 0\\ 0 \end{array}} \right]\] Therefore, the columns of chosen matrix span $R^3$
\[\left[ {\begin{array}{*{20}{l}} 0&0&2\\ 0&2&0\\ 2&0&0 \end{array}} \right]\]