(pp. 276-312)

For\[\text{A}=\left( \begin{array}{*{35}{r}} 3 & -3 & -1 & 7 & 7 & -2 \\ 13 & 14 & 12 & 10 & 7 & 0 \\ 2 & 14 & 10 & 11 & 11 & 6 \\ 9 & -2 & 2 & 4 & 14 & -4 \\ \end{array} \right),\]\[\text{B}=\left( \begin{array}{*{35}{r}} 2 & 4 & -2 & -7 \\ -3 & 3 & 1 & -5 \\ 6 & 10 & 1 & -8 \\ 3 & 4 & 4 & 6 \\ 9 & 9 & 5 & 12 \\ \end{array} \right),\]and\[\text{C}=\left( \begin{array}{*{35}{r}} 2 & 0 & 3 & 2 \\ 1 & 5 & -1 & 6 \\ 2 & 1 & 4 & 6 \\ \end{array} \right),\]please provide

(a)*M*,*N*

(b) rank*q*; is the matrix full rank? rank deficient?

(c) U and all necessary E_{i}s is

(d) basis$[{\cal R}(\cdot )]\subseteq {{\mathbb{R}}^{M}}$

(e) basis$[{\cal N}({{\cdot }^{T}})]\subseteq {{\mathbb{R}}^{M}}$

(f) basis$[{\cal R}({{\cdot }^{T}})]\subseteq {{\mathbb{R}}^{N}}$

(g) basis$[({\cal N}(\cdot )]\subseteq {{\mathbb{R}}^{N}}$

and show that

(h)${\cal R}(\cdot )\bot {\cal N}({{\cdot }^{T}})$

(i)${\cal R}({{\cdot }^{T}})\bot {\cal N}(\cdot )$

(j)${\cal R}(\cdot )+{\cal N}({{\cdot }^{T}})={{\mathbb{R}}^{M}}$

(k)${\cal R}({{\cdot }^{T}})+{\cal N}(\cdot )={{\mathbb{R}}^{N}}$

In many of the tests below, we need to distinguish values that are practically but not identically zero. For this purpose, format(ʹlongʹ) is useful, as it provides a far better numerical resolution, e.g.,

>> format(ʹlongʹ); disp(2/3)

0.666666666666667

>> format(ʹshortʹ); disp(2/3)

0.6667

(a) To get*M*and*N*, use [Ma, Na=size(A), [Mb, Nb=size...