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Section 6.3 Nullspaces, Row Spaces and Column Spaces: Homework
Exercises Exercises
Exercise Group.
For each of the following matrices:
Find a basis for \(N(A)\) (if possible; if not indicate why)
Give the dimension of \(N(A)\)
Write \(N(A)\) in element-form notation
Find a basis for row\((A)\)
Give the dimension of row\((A)\)
Write row\((A)\) in element-form notation
Find a basis for col\((A)\)
Give the dimension of col\((A)\)
Write col\((A)\) in element-form notation
Give the rank of \(A\)
1.
\(A=\left(\begin{array}{rrr}2\amp -3\amp -5\\4\amp -6\amp -10\\8\amp -12\amp -20 \end{array} \right)\)
2.
\(A=\left(\begin{array}{rrr}1\amp -1\amp 7\\3\amp 0\amp -5\\-4\amp 1\amp 16 \end{array} \right)\)
3.
\(A=\left(\begin{array}{rrr}1\amp -2\amp 5\\2\amp -3\amp -9\\-4\amp 0\amp 5 \end{array} \right)\)
4.
\(A=\left(\begin{array}{rrrr}1\amp 5\amp 9\amp 0\\2\amp -2\amp 6\amp 1\\-1\amp 21\amp -33\amp 2 \end{array} \right)\)
5.
\(A=\left(\begin{array}{rrrr}1\amp 1\amp 2\amp 4\\1\amp 2\amp 2\amp 5\\1\amp 3\amp 2\amp 6 \end{array} \right)\)
6.
\(A=\left(\begin{array}{rrr}1\amp 5\amp -7\\-7\amp 3\amp 12\\-4\amp 12\amp 0\\-3\amp 2\amp 0 \end{array} \right)\)
7.
\(A=\left(\begin{array}{rrr}3\amp 2\amp -1\\2\amp -3\amp 1\\1\amp 1\amp 6\\2\amp -2\amp 3 \end{array} \right)\)
8.
\(\A=\left(\begin{array}{rrr}1\amp -2\amp 2\\7\amp -4\amp 1\\3\amp 14\amp -20 \end{array} \right)\)
9.
\(\A=\left(\begin{array}{rrr}1\amp -3\amp 2\\5\amp -4\amp 2\\-4\amp -10\amp 8 \end{array} \right)\)
10.
\(\A=\left(\begin{array}{rrr}1\amp -2\amp 2\\7\amp -4\amp 1\\3\amp 14\amp -2 \end{array} \right)\)
11.
\(\A=\left(\begin{array}{rrr}1\amp -3\amp 2\\5\amp -4\amp 2\\-4\amp -10\amp 9 \end{array} \right)\)
12.
\(\A=\left(\begin{array}{rrrr}1\amp 2\amp -1\amp 1\\6\amp 4\amp -3\amp 1\\2\amp -4\amp 1\amp -3 \end{array} \right)\)
13.
\(\A=\left(\begin{array}{rrrrr}3\amp 4\amp -2\amp 1\amp 2\\6\amp 8\amp -1\amp -5\amp 1 \end{array} \right)\)
14.
\(\A=\left(\begin{array}{rrrr}1\amp 0\amp 1\amp 2\\3\amp 1\amp 5\amp 1\\-1\amp 1\amp 1\amp -7 \end{array} \right)\)
15.
\(\A=\left(\begin{array}{rrr}-2\amp 4\amp 1\\2\amp -3\amp 1\\-6\amp 7\amp 1\\4\amp 1\amp -1 \end{array} \right)\)
16.
\(\A=\left(\begin{array}{rrr}1\amp -1\amp 0\\-2\amp -6\amp 1\\3\amp 1\amp 1\\2\amp 0\amp -3 \end{array} \right)\)
17.
18.
19.
20.
