Question

Let T1 : R3 −→ R2 be the linear transformation given by the matrix
1 3 2
2 0 −1
and let T2 : R2 −→ R1 be the linear transformation given by the matrix [5 3].

(a) The kernel of T1 is one dimensional. Find a vector u1 which spans Ker(T1) (i.e., a basis for Ker(T1)).

Answer 1

that is, you want to find the solution to the system
1x + 3y + 2z = 0
2x + 0y +-1z = 0
because that's what it means for the a column vector (x y z)^t to be in the kernel of T1.