Description
1. Consider the space V of all vectors
{v = (v1, …, vn) ∈ R
n
such that v = ∇f(0) = (∂1f(0), ∂2f(0), …, ∂nf(0))
for some C
1
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 = R
n
.
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 =
x1
x2
x3
!
| a1x1 + a2x2 + a3x3 = 0)
, (1)
Y =
(
x =
x1
x2
x3
!
| b1x1 + b2x2 + b3x3 = 0)
, (2)
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? Justify
rigorously your answers. (1) {x ∈ R
n
: Ax = 0} ⊆ R
n
, 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 R
n
.
1 MA 502
5. In the following, determine the dimension of each subspace and find a
basis for it.
(1)
x = (x1, x2) | x1 + x3 = 0
⊆ R × R.
(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.
2 MA 502
