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Section 6.2 Nullspaces, Row and Column Spaces: In-Class Practice
Worksheet 6.2.1 Worksheet
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.
\(\A=\left(\begin{array}{rrr}6\amp -2\amp 3\\4\amp -1\amp 7 \end{array} \right)\)2.
\(\A=\left(\begin{array}{rrrr}1\amp -4\amp 3\amp 5\\4\amp -15\amp 0\amp 7 \end{array} \right)\)3.
\(\A=\left(\begin{array}{rr}5\amp 2 \end{array} \right)\)4.
\(\A=\left(\begin{array}{r}4\\5\\-2 \end{array} \right)\)5.
\(\A=\left(\begin{array}{rrr}5\amp 2\amp 0\\15\amp 6\amp 9\\-10\amp -4\amp 1\\-5\amp -2\amp 8 \end{array} \right)\)6.
\(\A=\left(\begin{array}{rrr}2\amp 4\amp -5\\5\amp 7\amp -2\\-1\amp -5\amp 4 \end{array} \right)\)7.
\(\A=\left(\begin{array}{rrrrrr}2\amp 3\amp 4\amp 2\amp 3\amp 4\\5\amp 6 \amp 7\amp 5\amp 6\amp 7\\8\amp 9\amp 10\amp 8\amp 9\amp 10\end{array} \right)\)8.
\(A=\left(\begin{array}{rrrr}1\amp -2\amp 3\amp 4\\-1\amp 3\amp 0\amp -6\\1\amp -1\amp 6\amp 2 \end{array} \right)\)9.
\(B=\left(\begin{array}{rrrr}1\amp -2\amp 3\amp 4\\-1\amp 3\amp 0\amp -6\\0\amp 1\amp 3\amp 1 \end{array} \right)\)
Exercise Group.
Let \(\vec{u}=\left( \begin{array}{r} 3 \\ 7 \\1 \end{array} \right)\text{.}\)
12.
Let \(\U\) be an upper-triangular matrix. State and prove: \(N(\U)=\left\{\vec{0}\right\}\) if and only if ___________________________.13.
Suppose \(N(\A)=\left\{\vec{0}\right\}.\)
Prove or disprove: \(N(\A^T\A)=\left\{\vec{0}\right\}.\)
Prove or disprove: \(N(\A\A^T)=\left\{\vec{0}\right\}.\)
14.
Prove Theorem 6.1.9.15.
Prove Corollary 6.1.41.
