Null, Column and Row Spaces
- Find Null space of A for the following
matrix. solution
\(A=\left[\begin{array}{rrrrr}1
& 5 & -4 & -3 & 1 \\ 0 & 1 & -2 & 1
& 0 \\ 0 & 0 & 0 & 0 &
0\end{array}\right]\)
- Find the matrix \(A\) where \(\mathrm{Col} \, A =
\left\{\left[\begin{array}{c}b-c \\ 2 b+c+d \\ 5 c-4 d \\
d\end{array}\right]: b, c, d \in \mathbb{R}\right\}\).
solution
- Let \(A=\left[\begin{array}{rrr}-8 & -2 & -9 \\ 6 &
4 & 8 \\ 4 & 0 & 4\end{array}\right]\) and
\(\overrightarrow{w}=\left[\begin{array}{r}2 \\ 1 \\ -2\end{array}\right]
\). Determine if \(\overrightarrow{w}\) is in \(\mathrm{Col} \, A
\). Is \(\overrightarrow{w}\) in \(\mathrm{Nul} \, A\) ? solution
- Find the bases for Null, Column and Row spaces of A. Given that \(B\) is row equivalent to \(A\)
where solution
\[
A=\left[\begin{array}{rrrrr}
-2 & -5 & 8 & 0 & -17 \\
1 & 3 & -5 & 1 & 5 \\
3 & 11 & -19 & 7 & 1 \\
1 & 7 & -13 & 5 & -3
\end{array}\right],
B=\left[\begin{array}{rrrrr}
1 & 3 & -5 & 1 & 5 \\
0 & 1 & -2 & 2 & -7 \\
0 & 0 & 0 & -4 & 20 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
\]
- Let \(M_{2 \times 2}\) be the vector space of all \(2 \times 2\) matrices, and define \(T: M_{2 \times 2} \rightarrow M_{2 \times 2}\) by \(T(A)=A+A^T\), where \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\).
a. Show that \(T\) is a linear transformation.
b. Let \(B\) be any element of \(M_{2 \times 2}\) such that \(B^T=B\). Find an \(A\) in \(M_{2 \times 2}\) such that \(T(A)=B\).
c. Show that the range of \(T\) is the set of \(B\) in \(M_{2 \times 2}\) with the property that \(B^T=B\).
d. Describe the kernel of \(T\).