PROBLEM SET II
DUE FRIDAY,  FEBRUARY
Exercise . Show that every nonzero vector space is, as a set, uncountable.
Exercise . Suppose θ ∈ [0, 2π), and consider the map Rθ : R2 . R2 such that for any vector v ∈ R2 , the vector
Rθ (v) is rotated by the angle θ. Show that Rθ is a linear isomorphism, and compute the corresponding matrix.
For any a nonnegative integer n, write Pn for the set of polynomials
a0 + a1 x + · · · + an xn ,
of degree at most n, where a0 , a1 , . . . , an ∈ R.
Exercise . Show that Pn defines a vector subspace of C0 (R), and show that the derivative d : C0 (R) .
restricts to a linear map d : Pn . Pn−1 .
C0 (R)
Exercise . Prove that the polynomials
{xk | 0 ≤ k ≤ n}
form a basis of Pn , whence Pn is (n + 1)-dimensional. Write the linear map d : Pn .
this basis.
Exercise . Prove that the polynomials
Pn−1 as a matrix relative to
{
}
1 k x 0 ≤ k ≤ n
k!
form a basis of Pn . Write the linear map d : Pn . Pn−1 as a matrix relative to this basis.
Exercise⋆ . Suppose {x0 , x1 , . . . , xn } a set of n + 1 distinct real numbers. Let In,k be the set {0, 1, . . . , n} − {k}.
For any k ∈ {0, 1, . . . , n}, write
∏ x − xm
pk (x) :=
.
xk − xm
m∈In,k
Prove that the polynomials
{
}
pk (x) 0 ≤ k ≤ n
form a basis of Pn . us for f ∈ Pn , we may write
f=
n
∑
αk pk
k=0
in a unique manner. Describe the coefficients αk .
Exercise⋆ . Suppose V a vector space, and suppose W ⊂ V a vector subspace. Show that there exists a linear map
φ : V . X with both of the following properties.
(.) For any v ∈ W, one has φ(v) = 0.
(.) For any linear map ψ : V . Y such that for any v ∈ W, one has ψ(v) = 0, there exists a unique linear map
η : X . Y such that ψ = η ◦ φ.
