suppose T: R^3 -> R^3 is given by T(x1,x2,x3) = (x1,-x2,0).

prove that T is a linear transformation. find the kernel, k, of T. Also, find dim K and dim (range T)

(Nod)

Printable View

- Dec 9th 2008, 12:21 PMpandakrapproving linear transformation in R^3
suppose T: R^3 -> R^3 is given by T(x1,x2,x3) = (x1,-x2,0).

prove that T is a linear transformation. find the kernel, k, of T. Also, find dim K and dim (range T)

(Nod) - Dec 9th 2008, 05:40 PMSkalkaz
Dear pandakrap,

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.

second condition...

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: $\displaystyle T = P_{xy}*R_y$ where $\displaystyle P_{xy}$ is a projection in xy-plane and $\displaystyle R_y$ is a reflection along y-axis. Why?