Practice Exam #2
Math 2270, Spring 2005
Problem 1. Let P2 be the space of all polynomials of degree ≤ 2 and consider the following two
bases:
U = (1, t, t2 )
and
B = (1, t − 1, (t − 1)2 ).
If T : P2 → P2 is the linear transformation given by T (f (t)) = f (2t − 1), find the following:
(a) The matrix A of T with respect to the basis U.
(b) The matrix B of T with respect to the basis B.
(c) The change of basis matrix S from the basis B to the basis U.
(d) Verify the formula SA = BS.
(e) Find the change of basis matrix from U to B.
(f) Is T an isomorphism?
Problem 2. Let V be the subspace of R3 given by V = span(v~1 , v~2 ) where
 


4
25
0 
v~1 =  0 
and
v~2 = 
3
−25
(a) Find an orthonormal basis U of V .


4
25
0 .
(b) Write a QR-factorization of the matrix A =  0
3 −25
(c) Find the matrix of the orthogonal projection on V with respect to the basis U.
(d) Find the matrix of the orthogonal projection on V with respect to the basis B = (v~1 , v~2 ) in
terms of the matrix A. Do not simplify!


49
 25 


~
~
(e) Find the least square solution of the system A~x = b, where b =  0 .
 7 
25
Problem 3. Prove that if W ⊆ V are two linear spaces, then dim(W ) ≤dim(V ).
Problem 4. Give the definition of the following:
(a) Orthogonal complement. Give an example.
(b) Orthogonal matrix. Give an example.
Problem 5. True or false? Justify all your answers.
(a) If A is an invertible matrix, then the equation (AT )−1 = (A−1 )T must hold.
(b) Every non-zero space has an orthonormal basis.
(c) Im(A)=Im(AT A) holds for all matrices A, n × m.
(d) The kernel of the linear transformation T (f ) = f ′ from P to P is {0}. (P is the set of all
polynomials.)