Math 110—Linear Algebra
Fall 2009, Haiman
Problem Set 3
Due Monday, Sept. 21 at the beginning of lecture.
1. Show that if vectors v1, . . . , vkin a vector space Vhave the properties that v16= 0, and
each viis not in the span of the preceding ones, then the vectors are linearly independent.
Conversely, show that if v1, . . . , vkis an ordered list of linearly independent vectors, then it
has the above properties.
2. (a) Find a formula for the number Q(n, k) of (ordered) sequences (v1, v2, . . . , vk) of linearly
independent vectors in V, where Vis a vector space of dimension nover F2, and k≤n.
[Hint: Use the previous problem and Problem Set 2, Problem 5.]
(b) Prove that the number of k-dimensional subspaces of (F2)nis given by
Q(n, k)/Q(k, k ), for k≤n.
(c) Calculate the number of 5-dimensional subspaces of (F2)10.
3. Let c1, . . . , cnbe distinct elements of a field F. Define the function E:Pm(F)→Fnby
E(f(x)) = (f(c1), . . . , f (cn)).
(a) Use Lagrange interpolation to prove that Eis onto if m≥n−1.
(b) Find the nullity of Eif m=n. Deduce that if f(x) is a polynomial of degree at most n
such that every ciis a root of f, then fmust be a scalar multiple of (x−c1)(x−c2)· · · (x−cn).
4. Let p(x)∈P(F) be a polynomial of degree d(exactly). Given n≥d, let Wbe the set of
polynomials in Pn(F) which are divisible by p(x).
(a) Prove that Wis a subspace of Pn(F).
(b) Find dim(W).
[Hint for both parts: show that Wis equal to the range of a linear transformation
T:Pn−d(F)→Pn(F) given by T(f(x)) = p(x)f(x).]
5. (a) Show that the function T:Cn→Cngiven by T((z1, . . . , zn)) = (z1, . . . , zn) is additive
(satisfies the first property in the definition of linear transformation) but not linear. Here z
denotes the complex conjugate a−bi of z=a+bi.
(b) Show that if we regard Cnas a vector space over Rinstead of C, then Tis linear.
6. Let S:Mm×n(F)→Fnbe the function that sends a matrix Ato the sum of its rows.
Assume mand nare non-zero.
(a) Prove that Sis a linear transformation.
(b) Find the range of S.
(c) Find the nullspace S.
(d) Let Nbe the set of matrices A∈Mm×n(F) such that every column of Asums to
zero. Use the preceding parts of this problem to prove that Nis a subspace of Mm×n(F),
and find its dimension.
7. Section 2.1 Exercise 17.