Description
1. Show that V = Z
2
5
is a vector space (with addition being modulo 5 and scalar multiplication also being modulo 5).
2. Let P
3 = {ax3 + bx2 + cx + d
a, b, c, d ∈ R} be the space of polynomials up to
degree 3 over the field R. Prove that P
3
is a vector space. (In other words, show that
vector addition and scalar multiplication is closed. Then, do your best to show the
other properties hold: associativity, additive identity, additive inverse, multiplicative
identity, and distributivity.)
To get you started, here is the proof of additive identity and the start of the proof for
additive inverses:
P
3 has an additive identity.
Proof. Consider 0x
3 + 0x
2 + 0x + 0, which we will write as 0P and call the “zero
polynomial.” Observe that 0P ∈ P
3
since 0 ∈ R. In addition, for any choice of
ax3 + bx2 + cx + d ∈ P
3
,
ax3 + bx2 + cx + d
+ 0P = (a + 0)x
3 + (b + 0)x
2 + (c + 0)x + (d + 0) (by vector addition)
= ax3 + bx2 + cx + d (by additive identity of the field)
Hence 0P is the additive identity of P
3
.
P
3
is closed under additive inverses.
Proof. Let ax3 + bx2 + cx + d ∈ P
3 be an arbitrary element in the space. Because
R is a field, there exists −a, −b, −c ∈ R such that a + (−a) = 0, b + (−b) = 0, and
c + (−c) = 0. · · · finish the argument