Question

Let the linear transformation defined by T(x,y,z) = (x –y +3z, 5x – 4y – 4z, 7x –6y+2z).
Find (i) Kernel(T), its basis and the dimension. (ii) Range (T), a basis and the dimension
of Range (T). Then verify the dimension theorem.

Answer 1

It may be easier to visualize as a matrix:

\[ T = \left(\begin{matrix}1 & -1 & 3 \\ 5& -4 & -4 \\ 7 & -6 & 2\end{matrix}\right)\]

Answer 2

From that it's a bit easier to find the kernel and range from known matrix techniques. For example, since the determinant is nonzero (unless I made a mistake) the columns are linearly independent and so the range has dimension 3.

Answer 3

Thanks