T is linear transformation if T(lambda*x) = lambda*T(x) and T(x+y)=T(x)+T(y).
Now the first codition:
Let x = (x1, x2, x3) so
T(lambda*x) = (lambda*x1, -lambda*x2, 0) = lambda*(x1, -x2, 0) = lambda*T(x) --> it's true.
y belongss to kernel if T(y) = 0.
Now let y = (y1, y2, y3)
T(y) = 0 --> (y1, -y2, 0) = (0, 0, 0)
so the condition of kernel is y1=0 and y2=0 --> Kernel(T): (0, 0, t) where t is real. --> the kernel is one-dimensional.
You know surely:
Dim(R^3) = Dim(Kernel(T)) + Dim(Range(T)) .
By the way, what is this T transformation doing? T affect as two linear transformation one after another: where is a projection in xy-plane and is a reflection along y-axis. Why?