1.
\(\left(\begin{array}{r}-2\\1\\5 \end{array} \right),\left(\begin{array}{r}0\\-3\\7 \end{array} \right)\in\) col\((A),\,\,\left(\begin{array}{r}1\\6\\-4 \end{array} \right)\in N(A)\)
Section 6.6 Constructing Matrices With Given Row, Column, or Nullspace Properties: Homework
Exercises Exercises
Exercise Group.
In each case, construct a matrix having the specified properties or explain why such a matrix does not exist. Be sure to include the plausibility/size check in your analysis.
2.
\(\left(\begin{array}{r}3\\2\\6 \end{array} \right),\left(\begin{array}{r}4\\1\\1 \end{array} \right)\in\) col\((A),\,\,\left(\begin{array}{r}-2\\8\\7 \end{array} \right)\in N(A)\)
3.
\(\left(\begin{array}{r}-2\\1\\5 \end{array} \right),\left(\begin{array}{r}0\\-3\\7 \end{array} \right)\in N(A),\,\left(\begin{array}{r}1\\6\\-4 \end{array} \right)\in\) col\((A)\)
4.
\(\left(\begin{array}{r}3\\2\\6 \end{array} \right),\left(\begin{array}{r}4\\1\\1 \end{array} \right)\in N(A)\text{,}\) \(\left(\begin{array}{r}-2\\8\\7 \end{array} \right)\in\) col\((A)\)
5.
\(\left(\begin{array}{r}2\\-2\\5 \end{array} \right)\in\) col\((A),\,\,\left(\begin{array}{r}8\\1\\5 \end{array} \right)\in N(A)\)
6.
\(\left(\begin{array}{r}3\\-2\\-8 \end{array} \right)\in\) col\((A),\,\,\left(\begin{array}{r}-2\\8\\7 \end{array} \right)\in N(A)\)
7.
\(N(A)=\text{ span } \left\{\left(\begin{array}{r}6\\2\\-3\\1 \end{array} \right)\right\}\)
8.
\(N(A)=\text{ span } \left\{\left(\begin{array}{r}-9\\3\\-4\\1 \end{array} \right)\right\}\)
9.
\(N(A)=\text{ span } \left\{\left(\begin{array}{r}6\\2\\-3\\1 \end{array} \right),\left(\begin{array}{r}5\\-2\\0\\3 \end{array} \right)\right\}\)
10.
\(N(A)=\text{ span } \left\{\left(\begin{array}{r}-9\\3\\-4\\1 \end{array} \right),\left(\begin{array}{r}3\\-7\\0\\4 \end{array} \right)\right\}\)
11.
col \((A)=\text{ span } \left\{\left(\begin{array}{r}2\\1\\1\\5 \end{array} \right),\,\left(\begin{array}{r}1\\1\\5\\2 \end{array} \right)\right\}\)
12.
col \((A)=\text{ span } \left\{\left(\begin{array}{r}-9\\3\\-4\\1 \end{array} \right),\,\left(\begin{array}{r}3\\-7\\0\\4 \end{array} \right)\right\}\)
13.
\(\left(\begin{array}{r}-5\\6\\12 \end{array} \right),\,\left(\begin{array}{r}3\\-3\\4 \end{array} \right)\) are contained in both the nullspace and the column space of \(A\)
14.
\(\left(\begin{array}{r}2\\1\\-3 \end{array} \right),\,\left(\begin{array}{r}4\\-4\\8 \end{array} \right)\) are contained in both the nullspace and the column space of \(A\)
15.
col \((A)\) contains \(\left(\begin{array}{r}1\\5\\-2 \end{array} \right),\,\left(\begin{array}{r}2\\0\\3 \end{array} \right)\) and whose nullspace consists of all multiples of (i.e. is spanned by) \(\left(\begin{array}{r}5\\-12\\8\\2 \end{array} \right)\text{.}\)
16.
col \((A)\) contains \(\left(\begin{array}{r}-2\\3\\-7 \end{array} \right),\,\left(\begin{array}{r}5\\6\\2 \end{array} \right)\) and whose nullspace consists of all multiples of (i.e. is spanned by) \(\left(\begin{array}{r}1\\-2\\5\\-2 \end{array} \right)\text{.}\)
17.
col \((A)\) contains \(\left(\begin{array}{r}-1\\0\\-2 \end{array} \right),\,\left(\begin{array}{r}3\\1\\4 \end{array} \right)\) and whose nullspace contains \(\left(\begin{array}{r}4\\-2 \end{array} \right)\text{.}\)
18.
col \((A)\) contains \(\left(\begin{array}{r}-2\\3\\-7 \end{array} \right),\,\left(\begin{array}{r}5\\6\\2 \end{array} \right)\) and whose nullspace contains \(\left(\begin{array}{r}1\\-3 \end{array} \right)\text{.}\)
19.
col \((A)=\left\{\left.a\left(\begin{array}{r}-1\\0\\-2 \end{array} \right)+b\left(\begin{array}{r}3\\1\\4 \end{array} \right)\right|\,\,a,b\in\R\right\}\text{,}\) \(N(A)=\left\{\left.a\left(\begin{array}{r}-3\\5\\3\\0 \end{array} \right)+b\left(\begin{array}{r}9\\2\\0\\4 \end{array} \right)\right|\,\,a,b\in\R\right\}\)
20.
col \((A)=\left\{\left.a\left(\begin{array}{r}-2\\3\\-7 \end{array} \right)+b\left(\begin{array}{r}5\\6\\2 \end{array} \right)\right|\,\,a,b\in\R\right\}\text{,}\) \(N(A)=\left\{\left.a\left(\begin{array}{r}5\\-2\\1\\0 \end{array} \right)+b\left(\begin{array}{r}4\\6\\0\\1 \end{array} \right)\right|\,\,a,b\in\R\right\}\)
