I want to solve the following task:
Find the minimal generating set (german: "minimales Erzeugendensystem") for the set S:
S = { $\begin{pmatrix} 1 \\ 1 \\ 0 \\1\\1 \end{pmatrix}$, $\begin{pmatrix} 0 \\ 0 \\ 1 \\1\\0 \end{pmatrix}$, $\begin{pmatrix} 0 \\ 1 \\ 0\\0\\0 \end{pmatrix}$, $\begin{pmatrix} 1 \\ 0\\ 0 \\1\\1 \end{pmatrix}$, $\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\1 \end{pmatrix}$ }
with S in $\mathbb{Q}^5$. A is the matrix with the vectors from S as its columns.
Usually I would just have used the gauss algorithm on the matrix A to achieve the reduced form with only vectors independent from each other left. However, the solution of the exercise looks totally different, and I don't understand what exactly is being done:
The solution says that with the the invertible Matrix W..
$$W =\begin{pmatrix} 1/2 & 0 & -1/2 & 1/2 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ -1/2 & 0 & 1/2 & 1/2 & 0 \\ 1/2 & 0 & 1/2 & -1/2 & 0 \\ -1 & 0 & 0 & 0 & 1 \end{pmatrix}.$$
...we arrive with the gauss algorithm at the equation:
W * A = $$\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1& 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}.$$
However it does not say from what we arrive at this matrix W, and why arriving at this equation is useful - and I can't really figure. Any help/hints? Many thanks
It certainly helps to show the system has rank $4$. I have no idea where $W$ comes from: this would require the context.
Furthermore, knowing $WA$ helps you extracting a minimal generating set from $S$. Indeed, it amounts to determine which vectors of $S$ can be expressed in function of the others. Note that left multiplying by an invertible matrix preserves linear relations between the column vectors of the matrix $A$, and conversely.
Let's denote $e_1, e_2,\dots,e_5$ the column vectors of $A$, and $e'_1,e'_2,\dots,e'_5$ those of $WA$. We clearly have $e'_5=e'_1+e'_2$, hence $e_5=e_1+e_2$.
Also $e'_1,e'_2,e'_3,e'_4$ are linearly independent, hence $e_1,e_2,e_3,e_4$ are.
Thus $\;\{e_1,e_2,e_3,e_4\}$ is a basis for $\langle\, S\,\rangle$.