91Ó°ÊÓ

Prove that (as noted at the beginning of this section) the range of a linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is the column space of its matrix \([T]\)

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{lll} a & 0 & 0 \\ 1 & a & 0 \\ 0 & 1 & a \end{array}\right]$$

Compute the rank and nullity of the given matrices over the indicated \(\mathbb{Z}_{p}\). \(\left[\begin{array}{cccc}1 & 3 & 1 & 4 \\ 2 & 3 & 0 & 1 \\ 1 & 0 & 4 & 0\end{array}\right]\) over \(\mathbb{Z}_{5}\)

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{lll} 0 & a & 0 \\ b & 0 & c \\ 0 & d & 0 \end{array}\right]$$

Compute the rank and nullity of the given matrices over the indicated \(\mathbb{Z}_{p}\). \(\left[\begin{array}{lllll}2 & 4 & 0 & 0 & 1 \\ 6 & 3 & 5 & 1 & 0 \\ 1 & 0 & 2 & 2 & 5 \\ 1 & 1 & 1 & 1 & 1\end{array}\right]\) over \(\mathbb{Z}_{7}\)

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{rrrr} 0 & -1 & 1 & 0 \\ 2 & 1 & 0 & 2 \\ 1 & -1 & 3 & 0 \\ 0 & 1 & 1 & -1 \end{array}\right]$$

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \end{array}\right]$$

If \(A\) is \(m \times n\), prove that every vector in null(A) is orthogonal to every vector in row \((A)\).

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$$

If \(A\) and \(B\) are \(n \times n\) matrices of rank \(n\), prove that \(A B\) has rank \(n\).