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 R 2 ?

What are all of the possible vector subspaces in R 2 ? A.The zero Vector B.Any line passing through the origin C. All of R 2

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)f8. 1f = f

Recall Subspace A subset W in R n is a subspace if it has the following 3 properties W contains the zero Vector in R n 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 R 3 ? A)The zero vector B) Any line passing through the origin C) Any plane containing the origin. D) All of R 3

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:x and –x 2 + x c) yes

Consider the elements f 1,f 2,f 3,…f n in a linear space V 1.We say that f 1,f 2,f 3,…f n span V if every f in V can be expressed as a linear combination of f 1,f 2,f 3,…f n 2. We say that f 1,f 2,f 3,…f n are linearly independent if the equation c 1 f 1 +c 2 f 2 +c 3 f 3 +…c n f n =0 has only the trivial solution where c 1 = … = c n = 0 3. We say that f 1,f 2,f 3,…f n 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=c 1 f 1 +c 2 f 2 +c 3 f 3 +…c n f n The coefficients c 1,c 2, …c n are called coordinates of f with respect to the basis β =(f 1,f 2,f 3,…f n ) 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, …x The dimension is n+1 23 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 f 1,f 2,f 3,…f n so that we can define its dimension dim(V) = n Otherwise, the space is called infinite dimensional Finite vs. Infinite Dimensionality

Homework p all 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.