1. Consider the space V of all vectors
{v = (v1,...,vn) ∈ Rn such that v = ∇f(0) = (∂1f(0),∂2f(0),...,∂nf(0)) for some C1 function f defined in a neighborhood of the origin}.
(1) Prove that V , equipped with the usual operations of vector sum and multiplication by a scalar is a vector space. (2) Prove that V = Rn. 2. Show that if X and Y are subspaces of a vector space V, then X ∩ Y is
also a subspace of V.
3. Consider
(
X = x =
,
(1)
(
Y = x =
,
(2)
x3
where the ai’s and bi’s are given real numbers.
(1) Prove that X and Y are vector spaces.
(2) Describe X ∩ Y in geometric terms, considering all possible choices of the coefficients. Is X ∩ Y a vector space?
4. Which of the following are subspaces of the given vector spaces? Justifyrigorously your answers. (1) {x ∈ Rn : Ax = 0} ⊆ Rn, where A is a given m × n matrix.
(2) {p ∈ P : p(x) = p(−x) for all x ∈ R} ⊆ P, where P is the set of all polynomials with real coefficients.
(3) {p ∈ P : p has degree less or equal than n} ⊆ P.
(4) {f ∈ C[0,1] : f(1) = 2f(0)} ⊆ C[0,1], where C[0,1] is the set of all continuous functions on [0,1]. (5) The unit sphere in Rn.
1
5. In the following, determine the dimension of each subspace and find abasis for it.
.
(2) The set of all n × n square matrices with real coefficients that are equal to their transpose.
(3){p ∈ P2 : p(0) = 0} ⊆ P2, where P2 is the set of all polynomials of degree ≤ 2.
