basis and dimension of kernel of T = (x + 2y - z, ....

Dear friends, How will I get basis and dimension of the kernel of T?

Let T:R^3---R^3 be linear mapping defined by T(x, y, z) = (x + 2y - z, y + z, x + y - 2z). Find the basis and dimension of the kernal of T.

Row reduce the transform matrix:
\(\displaystyle reff\left[ {\begin{array}{rrr}
1 & 2 & { - 1} \\
0 & 1 & 1 \\
1 & 1 & { - 2} \\
\end{array}} \right] = \left[ {\begin{array}{rrr}
1 & 0 & { - 3} \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{array}} \right].\)

From that we see that \(\displaystyle \left[ {\begin{array}{c}
{3\alpha } \\
{ - \alpha } \\
\alpha \\
\end{array}} \right]\) will be a basis of the null space for each alpha.