EITHER

The linear transformation \(T:\mathbb R^4\to\mathbb R^4\) is represented by

\(\mathbf M=\begin{pmatrix}1&2&3&4\\1&-1&2&3\\1&-3&3&5\\1&4&2&2\end{pmatrix}\).

The range space of \(T\) is denoted by \(V\).

(i) Determine the dimension of \(V\).

(ii) Show that the vectors \(\begin{pmatrix}1\\1\\1\\1\end{pmatrix}\), \(\begin{pmatrix}2\\-1\\-3\\4\end{pmatrix}\), and \(\begin{pmatrix}3\\2\\3\\2\end{pmatrix}\) are a basis of \(V\).

The set of elements of \(\mathbb R^4\) which do not belong to \(V\) is denoted by \(W\).

(iii) State, with a reason, whether \(W\) is a vector space.

(iv) Show that if \(\begin{pmatrix}x\\y\\z\\t\end{pmatrix}\) belongs to \(W\), then \(x+y\ne z+t\).