Determine the dimension of the subspace $W$ of $\mathbb{R}^{5}$ spanned by the vectors

$$\left(\begin{array}{r} 1 \\ 2 \\ 2 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{r} 4 \\ 2 \\ -2 \\ 6 \\ -2 \end{array}\right),\left(\begin{array}{l} 4 \\ 5 \\ 3 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 5 \\ 4 \\ 0 \\ 5 \\ -1 \end{array}\right)$$

Write down a $5 \times 5$ matrix $M$ which defines a linear map $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ whose image is $W$ and which contains $(1,1,1,1,1)^{T}$ in its kernel. What is the dimension of the space of all linear maps $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ with $(1,1,1,1,1)^{T}$ in the kernel, and image contained in $W$ ?