Find basis for the kernal and image of the linear transforma

Find basis for the kernal and image of the linear transformation T defined by T (x y z) = (4y -5y - 3z -5y - 3z 7z - 2y). Kernel basis: Image basis

Solution

We know that Ker(T) is the set of solutions to the ewquation T(X) = 0. If X = (x,y,z)^T, then this equation is equivalent to (4y,-5y-3z, -5y-3z, 7z-2y)^T = (0,0,0,0)^T or, 4y = 0, -5y-3z = 0 and 7z-2y = 0. Then x is arbitrary and y = 0,z = 0 so that X = (x,0,0)^T = x(1,0,0)^T. Hence a basis forKer(T) = {(1,0,0)^T}.

The image of a T consists of all the values that T takes in its codomain i.e. R⁴ which are of the form (4y,-5y-3z, -5y-3z, 7z-2y)^T = y(4, -5,,-5,-2)^T +z (0,-3,-3,7)^T. Hence a bais for Im(T) is {(4, -5,,-5,-2)^T, (0,-3,-3,7)^T}.