Question

determine the rank and nullity of the linear transformation T;R^3-R^4 defined by
T(x,y,z)=(4z,2x-y,-2x+y+5z)

Answer 1

Write down the matrix representation of T, call it M, and then row reduce it.
The row rank of that reduced matrix = row rank of the M = rank of T

The nullity of M = (# columns of M) - (row rank of M)

Answer 2

btw, check your definition of T. What you wrote down has the co-domain of T being a 3-dim space, not a 4-dim space.

Answer 3

where u looking for strobe?