Math 311-503 Spring 2007
Sample problems for Test 2
Any problem may be altered or replaced by a different one!
Problem 1 (20 pts.) Let P2be the vector space of all polynomials (with real coefficients)
of degree at most 2. Determine which of the following subsets of P2are vector subspaces. Briefly
explain.
(i) The set S1of polynomials p(x)∈ P2such that p(0) = 0.
(ii) The set S2of polynomials p(x)∈ P2such that p(0) = 0 and p(1) = 0.
(iii) The set S3of polynomials p(x)∈ P2such that p(0) = 0 or p(1) = 0.
(iv) The set S4of polynomials p(x)∈ P2such that (p(0))2+ 2(p(1))2+ (p(2))2= 0.
Problem 2 (20 pts.) Let Lbe the linear operator on R2given by
Lx
y=2−1
−3 2x
y.
Find the matrix of the operator Lrelative to the basis v1= (1,1), v2= (1,−1).
Problem 3 (30 pts.) Consider a linear operator f:R3→R3,f(x) = Ax, where
A=
5 3 5
2 1 2
1 0 1
.
(i) Find a basis for the image of f.
(ii) Find a basis for the null-space of f.
Problem 4 (30 pts.) Let B=
1 1 1
1 1 1
0 0 1
.
(i) Find all eigenvalues of the matrix B.
(ii) For each eigenvalue of B, find an associated eigenvector.
(iii) Is there a basis for R3consisting of eigenvectors of B?
Bonus Problem 5 (25 pts.) Let f1, f2, f3,... be the Fibonacci numbers defined by
f1=f2= 1, fn=fn−1+fn−2for n≥3. Find lim
n→∞
fn+1
fn
.
