Lecture 8
Chapter 4: Vector Spaces Vector Spaces
What is a vector space?
A set V together with an operation of addition,
usually denoted x + y for any x, y V , and an
operation of scalar multiplication, usually denoted
sx for any x V and s R, such that for any x, y, z
V and s, t, R we have all of the following
properties: (textbook, page 197)
2 V1 x + y V (closed under addition)
V2 (x + y) + z = x + (y + z) (addition is associative)
V3 There is an element 0 V, called the zero vector, such that
x + 0 = x = 0 + x (additive identity)
V4 For each x V there exists an element –x V such that
x + (-x) = 0 (additive inverse)
V5 x + y = y + x (addition is commutative
V6 sx V (closed under scalar multiplication)
V7 s(tx) = (st)x (scalar multiplication is associative)
V8 (s + t) x = sx + tx (scalar addition is distributive)
V9 s(x + y) = sx + sy (scalar multiplication is distributive)
V10 1x = x (scalar multiplicative identity)
3 Example 1
Let P 2enote the set of all real polynomials of
degree 2 or less. V erify that P2is a real vector
space.
4 Solution
Natural addition is associated with polynomials.
For example, let p(x) and q(x) be the
polynomials
p(x) = 2x – x + 3 and q(x) = x + 2x – 1
Need to get the sum r(x) = p(x) + q(x)
=> r(x) = 3x + x + 2
5 Scalar multiplication: Let s(x) = 2q(x)
s(x) = 2x + 4x – 2
Seems reasonable to expect P t2 be a real
vector space. Need to prove this more
rigorously.
6 Define P to2be set of all expressions (or
functions) of the form:
p(x) = a x + a x + a
2 1 0
Where a , 2 , 1nd a are0real constants.
7 Define the following two vectors in P : 2
2 2
p(x) = a x2+ a x + 1 and q(0) = b x + b x + 2 1 0
r(x) = p(x) + q(x) = (a + b )x + (a + b )x + (a + b )
2 2 1 1 0 0
Define the scalar multiple s(x) = cp(x)
s(x) = (ca 2x + (ca )x1+ (ca ) 0
8 Subspaces (textbook, page 201)

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