Consider T P2 rightarrow R2 defined by Ta bx cx2 a b Dete

Consider T: P_2 rightarrow R^2 defined by T(a + bx + cx^2) = [a b]. Determine ker T. Find a basis for ker T. Determine im T. Find a basis for im T.

Solution

The kernel of the linear transformation T is the set of all the solutions of the equation T( P₂(x) ) = 0. Let     a +bx +cx² be an arbitrary element of P₂(x), where a, b, c R are arbitrary. Now, T( P₂(x) ) = 0 if (a,b)^T = 0, i.e. if a = 0 and b = 0. Then all the elements of P₂(x) for which T( P₂(x) ) = 0 are of the form cx² where c R is arbitrary. Thus, a basis for the kernel of T is {x²}.

The image of the linear transformation T consists of all the values   that T takes in its co-domain. f is a Since T is a linear transformation from P₂ to R² , hence the image of T or, Im (T) = {T(u) ): u P₂} ={v R² :   v = T(u), for some u P₂ }. Now, an arbitrary element of R² is of the form (a,b)^T where a, b R are arbitrary. Also (a,b)^T    R² is the image of a +bx +cx² where a, b, c R are arbitrary. Further, (a, b)^T = a(1,0)^T =b(0,1)^T . Thus, Im(T) = { (a,b)^T : a, b R }. Also, a basis for Im(T) is { (1,0)^T , (0, 1)^T } as (a,b)^T being equal to a(1, 0)^T + b(0, 1)^T is a linear combination of (1,0)^T , (0, 1)^T for all a, b R.