Range space basis and dimension

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1 \\ 1 & 3 & - 1 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by \(V\).

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 2 \\ 3 \\ 0 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \\ - 4 \end{array} \right)\) are linearly independent.
3. Write down a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
4. State, with a reason, whether \(W\) is a vector space.
5. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(y - z - t \neq 0\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 1 & 5 & 7 \\ 3 & 9 & 17 & 25 \\ 1 & 7 & 7 & 11 \\ 3 & 6 & 16 & 23 \end{array} \right)\).

1. In either order,
   (a) show that the dimension of \(R\), the range space of T , is equal to 2 ,
   (b) obtain a basis for \(R\).
2. Show that the vector \(\left( \begin{array} { r } 1 \\ - 15 \\ - 17 \\ - 6 \end{array} \right)\) belongs to \(R\).
3. It is given that \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for the null space of T , where \(\mathbf { e } _ { 1 } = \left( \begin{array} { r } 14 \\ 1 \\ - 3 \\ 0 \end{array} \right)\) and \(\mathbf { e } _ { 2 } = \left( \begin{array} { r } 19 \\ 2 \\ 0 \\ - 3 \end{array} \right)\). Show that, for all \(\lambda\) and \(\mu\), $$\mathbf { x } = \left( \begin{array} { r } 4 \\ - 3 \\ 0 \\ 0 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ is a solution of $$\mathbf { M x } = \left( \begin{array} { r } 1 \\ - 15 \\ - 17 \\ - 6 \end{array} \right)$$
4. Hence find a solution of \(( * )\) of the form \(\left( \begin{array} { c } \alpha \\ 0 \\ \gamma \\ \delta \end{array} \right)\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { r r r r } - 1 & 2 & 3 & 4 \\ 1 & 0 & 1 & - 1 \\ 1 & - 2 & - 3 & a \\ 1 & 2 & 5 & 2 \end{array} \right) .$$

1. For \(a \neq - 4\), the range space of T is denoted by \(V\).
   (a) Find the dimension of \(V\) and show that $$\left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \quad \left( \begin{array} { r } 2 \\ 0 \\ - 2 \\ 2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4 \\ - 1 \\ a \\ 2 \end{array} \right)$$ form a basis for \(V\).
   (b) Show that if \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(V\) then \(x + 2 y = t\).
2. For \(a = - 4\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right)$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 2 & - 1 & \alpha \\ 2 & 3 & - 1 & 0 \\ 2 & 1 & 2 & - 2 \\ 0 & 1 & - 3 & - 2 \end{array} \right)$$ Given that the dimension of the range space of T is 4 , show that \(\alpha \neq 1\). It is now given that \(\alpha = 1\). Show that the vectors $$\left( \begin{array} { l } 1 \\ 2 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 3 \\ 1 \\ 1 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 2 \\ - 3 \end{array} \right)$$ form a basis for the range space of T . Given also that the vector \(\left( \begin{array} { c } p \\ 1 \\ 1 \\ q \end{array} \right)\) is in the range space of T , find a condition satisfied by \(p\) and \(q\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4 \\ 3 & 4 & 6 & 1 \\ - 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,

1. show that the dimension of \(R\) is 2 ,
2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right) .$$

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 3 & 4 \\ 1 & - 1 & 2 & 3 \\ 1 & - 3 & 3 & 5 \\ 1 & 4 & 2 & 2 \end{array} \right)$$ The range space of T is denoted by \(V\).

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \\ 4 \end{array} \right) , \left( \begin{array} { l } 3 \\ 2 \\ 3 \\ 2 \end{array} \right)\) are a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
3. State, with a reason, whether \(W\) is a vector space.
4. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4 \\ 3 & 4 & 6 & 1 \\ - 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,

1. show that the dimension of \(R\) is 2 ,
2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)$$