4.1 Introduction to Linear Spaces(a.k.a. Vector Spaces)

Recall: A SubspaceA subspace of a linear space V is called a

subspace if:a) W contains the neutral element 0 of Vb) W is closed under additionc) 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 VectorB. Any line passing through the originC. All of R2

Linear Spaces aka Vector SpacesA 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 Property2. f+g=g+f Commutative Property3. There exists a neutral element such that f+n =fThis n is unique and denoted by 04. For each f in V there exists g such that f+g=05. k(f+g) =kf+kg Distributive Property6.(c+k)f = cf + kf, Distributive Property7.c(kf) = (ck)f 8. 1f = f

RecallSubspace

• 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 vectorB) Any line passing through the originC) 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) yesb) No, not closed under addition Example:x2 + 3 and –x2 + xc) 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

1, x, x , x , …xThe dimension is n+1

2 3 n

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

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

Finite vs. Infinite Dimensionality

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.