Consider the linear transformation L : R^3 --> R^5 given by L(~x) = A~x where A

is the matrix

A = 1 2 1

2 alpha beta

1 beta 1

2 alpha beta

-1 -2 -1

a) Find dim(ker(L)) as a function of alpha and beta. For what values of alpha, beta; is this as large as possible? For what values of alpha, beta; is this as small as possible?

b) Find dim(Range(L)) as a function of alpha and beta. For what values of alpha, beta; is this as large as possible? For what values of alpha, beta; is this as small as possible?

c) Now consider the vectors

v1 = 1 v2 = 1 v3 = 1

1 1 b

1 a b

What conditions on a; b ensure that these three vectors form a basis?