Problem A.
1. The linear map T : R3 -> R4 is given by the matrix
with respect to the standard bases in R3 and in R4. Find bases of Ker T and ImT.
2. Let S denote the set of polynomials in R[z] that have degree at most 5 and root 3/2 . Is S a vector space over R? Justify your answer.
In the case of armative answer compute the dimension of S.
3. (a) Show that C = {z = x + y√-1 : x; y ε R} is a vector space of dimension 2 over R.
(b) Define a linear map T from the vector space C to the vector space Mat(2; 2;R) of 2 2 matrices by the rule
Compute dim ImT.
(c) Show that T(z1z2) = T(z1)T(z2), where z1z2 is a usual multiplication of two complex numbers z1 and z2, and T(z1)T(z2) is the matrix multiplication of the two corresponding matrices.
4. Suppose linear maps T 2 L(V;W) and S 2 L(W; U) of finite-dimensional vector spaces are given. Show that
5. Show that the set of invertible n x n matrices (that is, the ones that correspond to invertible linear maps from Fn to Fn) is a group with respect to the matrix multiplication and with the identity matrix I (which has ones along the main diagonal and zeroes otherwise).