Dimension and Rank of Vector Spaces
1. Find a basis, and state the dimension of the subspace
\(
H=\left\{\left[\begin{array}{c}
s+t\\
2s \\
3s-t \\
-t
\end{array}\right]: s, t\text { in } \mathbb{R}\right\} \). solution
2. Find a basis, and state the dimension of the subspace
\(
H=\left\{\left[\begin{array}{c}
a-3 b+6 c \\
5 a+4 d \\
b-2 c-d \\
3 d
\end{array}\right]: a, b, c, d \in \mathbb{R}\right\}
\). solution
3. Determine the dimension of null space and rank for each of the following matrices. solution
\(A=\left[\begin{array}{rrrrr}1 & -6 & 9 & 0 & -2 \\ 0 & 1 & 2 & -4 & 5 \\ 0 & 0 & 0 & 5 & 1 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)
\(B=\left[\begin{array}{rrrrrr}1 & 3 & -4 & 2 & -1 & 6 \\ 0 & 0 & 1 & -3 & 7 & 0 \\ 0 & 0 & 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\)
\(C=\left[\begin{array}{rrrr}1 & 0 & 9 & 5 \\ 0 & 0 & 1 & -4\end{array}\right]\)
4. Assume that the matrix A is row equivalent to B. Without calculations, list rank A and dim Nul A. Then find bases for Col A, Row A, and Nul A. solution
\( A=\left[\begin{array}{rrrrr}1 & -3 & 4 & -1 & 9 \\ -2 & 6 & -6 & -1 & -10 \\ -3 & 9 & -6 & -6 & -3 \\ 3 & -9 & 4 & 9 & 0\end{array}\right], \quad B=\left[\begin{array}{rrrrr}1 & -3 & 0 & 5 & -7 \\ 0 & 0 & 2 & -3 & 8 \\ 0 & 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)