Rank , Kernel and Other Properties ofa Linear Transformation

Question

Determine rank and kernel of L : P2 --> M2*2(R) given by,

L(ax^2 + bx + c) = A, where A is 2*2 matrix whose elements are as follows : A11 = A12 = a & A21 = A22 = c

Answer 1

We need to find the dimension of the image of L since it is equal to the rank of L. Note that

L(ax^2 + bx +c) = [a, a; c c ]= a*[1, 1; 0, 0] + c*[0,0;1,1], that is, the image of L is contained in the span of

{[1,1; 0, 0], [0,0;1,1]}.

On the other side, for any matrix [a,a; c,c] in the span of {[1,1; 0, 0], [0,0;1,1]}, it is the image of ax^2+c under L, that is, the span of {[1,1; 0, 0], [0,0;1,1]} is contained in the image of L. Now we can conclude that the image of L is equal to the span of {[1,1; 0, 0], [0,0;1,1]}. Furthermore, since [1,1; 0, 0] and [0,0;1,1] are linearly independent, then the dimension of the span of {[1,1; 0, 0], [0,0;1,1]} is 2. As a result, the dimension of the image of L is also 2 and so the rank of L is 2.

The kernel of L is ker(L)={ax^2+bx+c|L(ax^2+bx+c)=[0,0; 0,0]}

={ax^2+bx+c | [a,a; c,c]=[0,0;0,0] }

={bx | b is any real number}

= the span of {x}.

So the kernel of L is equal to the span of {x}.