Here are some problems to practice in order to prepare for the second oral exam. You may send me solutions. I will review your solutions but no grade will be given. I'll be happy to answer your questions.

1. Consider the map T from C[0,1] to M_2,2 (the vector space of all 2 by 2 matrices) which takes every function f(x) to the matrix

a) Prove that it is a linear transformation.

b) What is the range and what is the kernel of it.

2. Change the matrix in the previous problem to

and answer the same questions a) and b).

3. Consider the map from C[0,1] to C[0,1] which takes every function f(x) to the integral from 0 to x of f(t) dt (this is one of the examples considered in the Notes). Does this transformation preserve norms? Recall that the norm of a function f(x) is sqrt(integral(f(x)² dx) from 0 to 1).

4. Consider the map from C[0,1] to C[0,1] which takes every function f(x) to the function -f(x).

a) Is it a linear transformation?

b) Does this transformation preserve norms?

c) What is the range and what is the kernel?