Define the linear transformation T by T(x) Ax. Find ker(), nullity(), range(T), and rank A = [4/3 -4/3 2/3
-4/3 4/3 -2/3
2/3 -2/3 1/3]
(a) ker(T)
STEP 1: The kernel of T is given by the solution to the equation T(x)-0. Let x = (x1, x2, x3) and find x such that T(x) = 0. (1f there are an infinite number of solutions use t and s as your parameters.)
STEP 2: Use your result from Step 1 to find the kernel of T. (If there are an infinite number of solutions use t and s as your parameters.) ker() (b) nullity()
STEP 3: Use the fact that nullity) dim(kerT)) to compute nullity ). (c) range(T)
STEP 4: Transpose A and find its equivalent reduced row-echelon form. 2 2 1
STEP 5: Use your result from Step 4 to find the range ofT 0 1(2t, -2t, t): t is real) O ((2t, -2t, t): t is real) O ((t, -t, t): t is real) (d) rank(T)
STEP 6: Use the fact that rank() dim(range(T)) to compute rank(T)