Find nucleus image range and nullity of the given linear tra

Find nucleus, image, range and nullity of the given linear transformation

x z 3, calcule los valores caracteristicos, los vectores caracteristicos y los espacios caracteristicos

Solution

The null space or the kernel of a linear transformation T, i.e. Ker(T) is the set of solutions to the equation T(X) = 0. If X = (x,y,z,w)^T, then this equation is equivalent to x+z = 0 or, x = -z and y+w = 0 i.e. y = -w. Then X = (-z,-w,z,w)^T = z(-1,0,1,0)^T +w(0,-1,0,1)^T. Hence, Ker(T) = span{(-1,0,1,0)^T,(0,-1,0,1)^T }. The nullity of T is the dimension of its null space or Kernel. Hence, the nullity of T is 2.

The image of a linear transformation T, i.e. Im (T) = { T(X) : X R⁴}. If X=(x,y,z,w)^T, thenT(X)=(x+z,y+w)^T = x(1,0)^T+z(1,0)^T+y(0,1)^T+w(0,1)^T. Hence Im(T) = span{ (1,0)^T, (0,1)^T}. The range of a linear transformation is the same as its image.