Download 4.1 Introduction to Linear Spaces

4.1 Introduction to Linear Spaces
(a.k.a. Vector Spaces)
Recall: A Subspace
A subspace of a linear space V is called a
subspace if:
a) W contains the neutral element 0 of V
b) W is closed under addition
c) W is closed under scalar multiplication
Recall: What are all of the possible
vector subspaces in R2?
What are all of the possible vector
subspaces in R2?
A. The zero Vector
B. Any line passing through the origin
C. All of R2
Linear Spaces aka Vector Spaces
A linear Space is a set with two well defined
operations, addition and scalar multiplication.
Here are the properties that must be satisfied
1. (f+g)+h = f+(g+h) Associative Property
2. f+g=g+f Commutative Property
3. There exists a neutral element such that f+n =f
This n is unique and denoted by 0
4. For each f in V there exists g such that f+g=0
5. k(f+g) =kf+kg Distributive Property
6.(c+k)f = cf + kf, Distributive Property
7.c(kf) = (ck)f
8. 1f = f
Recall
Subspace
•
•
•
•
A subset W in Rn is a subspace if it has
the following 3 properties
W contains the zero Vector in Rn
W is closed under addition (of two vectors
are in W then their sum is in W)
W is closed under scalar multiplication
Example 11
Show that the differentiable functions form a
a subspace
Example 11 Solution
What are all of the vector subspaces
of R3?
A) The zero vector
B) Any line passing through the origin
C) Any plane containing the origin.
D) All of R3
Example 12
a) Is the set of all polynomials a subspace?
b) Is the set of all polynomials of degree n a
subspace?
c) Is the set of all polynomials with
degree < n a subspace?
Solution to 12
a) yes
b) No, not closed under addition
Example:x2 + 3 and –x2 + x
c) yes
Consider the elements f1,f2,f3,…fn in a linear space V
1. We say that f1,f2,f3,…fn span V if every f in V can be
expressed as a linear combination of f1,f2,f3,…fn
2. We say that f1,f2,f3,…fn are linearly independent if the
equation c1f1+c2f2+c3f3+…cnfn =0 has only the trivial
solution where c1= … = cn = 0
3. We say that f1,f2,f3,…fn are a basis for V if they are both
linearly independent and span V that means that every f in
V can be written as a linear combination of
f=c1f1+c2f2+c3f3+…cnfn
The coefficients c1,c2, …cn are called coordinates of f with
respect to the basis β =(f1,f2,f3,…fn )
The vector
is called the coordinate vector of f denoted
by [f]β
Dimension
If a linear Space has a basis with n elements
then , all of the other basis consist of n
elements as well. We say that n is the
dimension of V or
dim(V) =n
Example 15
Example 15 Solution
Coordinates
Finding a basis of a linear Space
1) A write down a typical element in terms of
some arbitrary constants
2) Using the arbitrary constants as
coefficients, express your typical element as
a linear combination of some elements of V.
3) Verify that all the elements of V in this
linear combination are linearly independent.
Example 16
Example 16 solution
Example
Find a basis and the dimension for all
polynomials of degree n or less
Example Solution
A basis would be
2
3
n
1, x, x , x , …x
The dimension is n+1
Find a basis for the set of all
polynomials What dimension is the
linear space containing the set of all
polynomials?
Note the answer is on the
next slide
Finite vs. Infinite Dimensionality
A linear Space V is called Finite dimensional
if has a (finite) basis f1,f2,f3,…fn so that we
can define its dimension dim(V) = n
Otherwise, the space is called infinite
dimensional
Homework p 163 1-16 all
17-41 odd
• Q: What is the physicist's definition of a
vector space?
• A: A set V such that for any x in V, x has a
little arrow drawn over it.