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Mobile App Chapter 1: Q39E (page 1)
Determine if the columns of the matrix span \({R^4}\).
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\{ - 9}&4&{ - 8}&7&{ - 3}\\{ - 6}&{11}&{ - 7}&3&{ - 9}\\4&{ - 6}&{10}&{ - 5}&{12}\end{array}} \right]\)
Short Answer
The columns of the matrix are in span \({R^4}\) .
Step by step solution
Solve matrix
Apply the Gaussian elimination method to solve the given matrix to find the pivot.
Apply row operation \({R_2} \to {R_2} + \frac{3}{4}{R_1}\).
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\{ - 6}&{11}&{ - 7}&3&{ - 9}\\4&{ - 6}&{10}&{ - 5}&{12}\end{array}} \right]\)
Now, apply row operation \({R_3} \to {R_3} + \frac{1}{2}{R_1}\) again in the above matrix.
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\0&{\frac{{15}}{2}}&{ - \frac{3}{2}}&{ - \frac{3}{2}}&{ - \frac{{13}}{2}}\\4&{ - 6}&{10}&{ - 5}&{12}\end{array}} \right]\)
Row operation in the matrix
Apply row operation \({R_4} \to {R_4} - \frac{1}{3}{R_1}\) in the above matrix.
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\0&{\frac{{15}}{2}}&{ - \frac{3}{2}}&{ - \frac{3}{2}}&{ - \frac{{13}}{2}}\\0&{ - \frac{{11}}{3}}&{\frac{{19}}{3}}&{ - 2}&{\frac{{31}}{3}}\end{array}} \right]\)
Apply row operation \({R_3} \to {R_3} + 6{R_2}\) in the above matrix.
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\0&0&0&0&{ - 2}\\0&{ - \frac{{11}}{3}}&{\frac{{19}}{3}}&{ - 2}&{\frac{{31}}{3}}\end{array}} \right]\)
Pivot of a matrix
Apply row operation \({R_4} \to {R_4} - \frac{{44}}{{15}}{R_2}\) in the above matrix to get the pivot of a matrix.
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\0&0&0&0&{ - 2}\\0&0&{\frac{{28}}{5}}&{ - \frac{{41}}{{15}}}&{\frac{{122}}{{15}}}\end{array}} \right]\)
Now, after interchanging the rows \({R_3} \leftrightarrow {R_4}\) in the above matrix, you get:
\(\left[ {\begin{array}{*{20}{c}}{12}&{ - 7}&{11}&{ - 9}&5\\0&{ - \frac{5}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{3}{4}}\\0&0&{\frac{{28}}{5}}&{ - \frac{{41}}{{15}}}&{\frac{{122}}{{15}}}\\0&0&0&0&{ - 2}\end{array}} \right]\)
Determine the span
The pivots in the matrix columns are represented as:
The matrix has a pivot in every row.
Hence, the columns of the matrix are in span \({R^4}\).
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