I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one.
W : $ x+y-z+u=1 $
$ 2x+u=2 $
$ z -u=0 $
I don't understand how I am supposed to get the parametric equation of that manifold. Can you help me please?
Generally, to be safe, write the linear system as $Ax=0$ then find a basis for $Null(A) = \{ x | Ax=0 \}$. The span of the basis is essentially the patch for the linear manifold.
For example, $x+y+z+u=0$ and $x-u-z=0$ so $A = \left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 0 & -1 & -1 \end{array} \right]$. We calculate, $$\text{rref}(A|0) = \left[ \begin{array}{cccc|c} 1 & 0 & -1 & -1 & 0 \\ 0 & 1 & 2 & 2 & 0\end{array} \right] $$ Traditionally, I use $z,u$ as parameters, but some insist we write $z=s$ and $u=t$ so I'll play along, note row one gives $x = z+u$ and row 2 gives $y=-2z-2u$ hence $$X(s,t) = ( s+t, -2s-2t, s, t) $$ This parametrizes the plane defined by the set of equations I gave.
Of course, for simple examples adding equations etc... wins, but, beware the problem of dependent equations. Row reduction is the natural enemy of the hidden clone equation.