Vector Spaces - 0929
1
Vector Spaces
Throughout this course, we work over R or C. We use k to denote the base field, i.e., k “ R or
k “ C. Usually n denotes an integer ě 0.
Elements of k are called scalars.
Definition. (Section 3.1) A vector space over k is an nonempty set V with an addition and
a scalar multiplication:
`:V ˆV ÑV
¨:kˆV ÑV
such that for any x, y, z P V and α, β P k, the following axioms hold:
1. x ` y “ y ` x.
2. px ` yq ` z “ x ` py ` zq.
3. There exists an element 0 P V such that 0 ` x “ x for all x P V .
4. For each x P V there exists an element w P V such that x ` w “ 0. This w is usually denoted
by p´xq, called the additive inverse of x.
5. α ¨ px ` yq “ α ¨ x ` α ¨ y.
6. pα ` βq ¨ x “ α ¨ x ` β ¨ x.
7. pαβqx “ αpβxq
8. 1 ¨ x “ x.
Example. k n : n-tuple of scalars, with coordinate-wise addition and scalar multiplication. More
precisely
k n “ tpx1 , ..., xn q : x1 , ..., xn P ku
Addition is defined as
px1 , ..., xn q ` px1 , ..., yn q “ px1 ` y1 , ..., xn ` yn q
Scalar multiplication is defined as
α ¨ px1 , ..., xn q “ pαx1 , ..., αxn q.
(We call them coordinate-wise addition and scalar multiplication).
We check Axiom 1 and Axiom 3 for k n .
Axiom 1. Suppose we have x, y P k n , we want to show that x ` y “ y ` x. Write
x “ px1 , ..., xn q
and
y “ py1 , ..., yn q
1
Then
x ` y “ px1 ` y1 , ..., xn ` yn q
and
y ` x “ py1 ` x1 , ..., yn ` xn q
At each coordinate, by commutativity of addition of real/complex numbers, we have
xi ` yi “ yi ` xi
Hence
x`y “y`x
as desired.
Axiom 3. We claim the zero element in k n is
p0, 0, ..., 0q.
We have
p0, 0, ..., 0q ` px1 , x2 , ..., xn q “ p0 ` x1 , 0 ` x2 , ..., 0 ` xn q “ px1 , x2 , ..., xn q.
Example. Pn : polynomials of degree ď n. Elements are in the form of
an X n ` an´1 X n´1 ... ` a1 X ` a0
with an , ..., a0 P k. Addition is the normal addition of polynomials, and scalar multiplication is the
standard multiplication as well.
2