1. ## Linear Mapping..

Determine whether there exist a linear map in the following case:

$\displaystyle T:V_3 \to V_3$ such that $\displaystyle T(0,1,2) = (3,1,2)$ and $\displaystyle T(1,1,1)=(2,2,2)$

If it exists give the general formula..

Pl help to solve this problem..

2. If $\displaystyle T$ is a linear transformation, then there exists a transformation matrix $\displaystyle A$ for the mapping such that $\displaystyle A\vec{x} = \vec{y}$. It's clear that such a matrix must be 3x3.

Let $\displaystyle A = \left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{ 22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]$

And use the equations you have. Multiply the $\displaystyle \vec{x}$ with A and equate them to $\displaystyle \vec{y}$. You should have infinite solutions to the problem

3. ## for existence...

For existence, will this work?
\displaystyle \left[ \begin {array}{ccc} 0&1&1\\ \noalign{\medskip} 1&1&0\\ \noalign{\medskip} 1&0&1 \end {array} \right]

4. Yes, that works fine. But the general forumla will be in terms of the a's of the matrix A

5. ## a little more general

Let's generalize the example a little. Will this work?
\displaystyle \left[ \begin {array}{ccc} a&1-2a&1+a\\ \noalign{\medskip} 1&1&0\\ \noalign{\medskip} 1&0&1 \end {array} \right]