Math 333 (2015) - Homework 3
Due: October 15, 2015.
NAME:
1. (10) State which of the following sets V are vector spaces. If it is not a vector space, state all the
axioms (A1)-(A10) that are not satisfied.
a) Matrix addition and scalar multiplication and
a
V = u ∈ M22 : u =
1
1
b
f or some a, b ∈ IR
b) The set V = IR2 where if x, y ∈ V and α ∈ IR addition and scalar multiplication are defined by:
x⊕y
=
(x1 y1 , x2 y2 )
αx
=
(αx1 , αx2 )
2. (20) Determine if each W below is closed under addition and scalar multiplication. Then state if W
is/is not a subspace of the associated vector space V .
a b
a) W = u ∈ M22 : u =
where a + b + c + d = 0
, V = M22
c d
b) W = {f ∈ C(IR) : f (x) ≤ 0}
, V = C(IR)
c) W = {f ∈ C(IR) : f (0) = 0} , V = C(IR)
d) W = A ∈ Mnn : AT = −A
, V = Mnn
3. (15) For each of the following express w as a linear combination of the vectors vk indicated. The vector
space V which w, vk are elements of are also indicated.
a) V = IR3 , w = (7, 8, 9), v1 = (2, 1, 4), v2 = (1, −1, 3), v3 = (3, 2, 5).
b) V = P2 , w = 5x2 + 13x + 3, v1 = x2 + 2x + 3, v2 = −x2 − 3x + 1.
−1 −5
1 2
2 1
c) V = M22 , w =
, v1 =
, v2 =
.
4
1
−1 0
1 1
4. (10) In the following problems compute the coordinate (w)S of w ∈ V relative the indicated bases. If,
for example,
S = {v1 , v2 , v3 }
where the vectors vk are defined then the coordinate (w)S is that vector c = (c1 , c2 , c3 ) ∈ IR3 such that
w = c1 v1 + c2 v2 + c3 v3 .
a) V = IR2 , v1 = (1, 2), v2 = (−1, 1), S = {v1 , v2 }, w = (1, 1)
b) V = P2 , v1 = x2 + 1, v2 = x + 1, v3 = x − 1,S = {v1 , v2 , v3 }, w = x2 + x + 1.
5. (5) Let S be the set of polynomials
S = {x3 , x2 + x, x3 + x2 + x, x3 − x}
and W = span(S). Determine a basis S 0 for W which contains only elements of S. What is dim(W )?