C 21 - Page 305 in paper copy of V2.02

Consider the subspace:

\[W = \langle \left\{ \left[\begin{array} {rr} 2 & 1 \\ 3 & -1 \end{array}\right] ,\left[\begin{array} {rr} 4 & 0 \\ 2 & 3 \end{array}\right] ,\left[\begin{array} {rr} -3 & 1 \\ 2 & 1 \end{array}\right] \right\} \rangle \]

of the vector space \(M_{22}\). Is

Solution: Convert to the augmented form:

\[C = \left[ \begin{array}{rr} -3 & 3\\ 6 & -4 \end{array}\right]\] an element of W?

\[C = \left[ \begin{array}{rrr|r} 2 & 4 & -3 & -3\\ 1 & 0 & 1 & 3\\ 3 & 2 & 2 & 6\\ -1 & 3 & 1 & -4 \end{array}\right]\]

Row Reduce: (using RREF in dMod package)

require(dMod)
A = matrix(c(2,1,3,-1,4,0,2,3,-3,1,2,1,-3,3,6,-4), nrow = 4)
rref(A, tol = sqrt(.Machine$double.eps), verbose = FALSE, fractions = TRUE)[1] #returning the reduced matrix only
## [[1]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    2
## [2,]    0    1    0   -1
## [3,]    0    0    1    1
## [4,]    0    0    0    0

Because it is possible to reduce these equations into appropriate RREF form, with pivots of 1 or 0 where i=j and all others zero on the left side of the equation, this set is a solution to the three matrices which are subsets of W and therefore ALSO is a subspace of W.