Chapter 3, Problem 3E

(Column Space arid Reduced Row Echelon Form)
Set
$B=\text{round}\left(10\ast \text{rand}\left(8,4\right)\right)$
$X=\text{round}\left(10\ast \text{rand}\left(4,3\right)\right)$
$C=B\ast X$
$A=\left[\begin{array}{cc}B& C\end{array}\right]$
(a) How are the column spaces of B and C related? (See Exercise 28 in Section 3.6.) What would you expect the rank of A to be? Explain. Use MATLAB to check your answer.
(b) Which column vectors of A should form a basis for its column space? Explain. If U is the reduced row echelon form of A. what would you expect its first four columns to be? Explain. What would you expect its last four rows to be? Explain. Use MATLAB to verify your answers by computing U.
(c) Use MATLAB to construct another matrix $D=\left[\begin{array}{cc}E& EY\end{array}\right]$ ,whereE is a random $6\times 4$ matrix and Y is a random $4\times 2$ matrix. What would you expect the reduced row echelon form of D to be? Compute it with MATLAB. Show that, in general, if B is an $m\times n$ matrix of rank n and X is an $n\times k$ matrix, the reduced row echelon form of $\left(\begin{array}{cc}B& BX\end{array}\right)$ will have block structure
$\left(\begin{array}{cc}1& X\end{array}\right)$ if $m=n$ or $\left(\begin{array}{cc}I& X\\ O& O\end{array}\right)$ if $m>n$