lesson 2 vector spaces

7/29/2019 Lesson 2 Vector Spaces

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Lesson2

Matrices: Vector spaces and linear

independencePP 9 - 14


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Vector spaces In Physics: north

In basic maths:

a vector in 2D

a vector in 3D

In general: is avector in n-space with every a realnumber

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a 3; 2

b 2; 1;5

1 2 nx a ,a ,...,a ai


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Notation : the set of all vectors in n- space

: the set of all vectors in the form

NB! Vector can refer to almost

anything a set of real numbers,

a set of complex numbers,

a set of matrices ect.

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Rn

2R ,a b


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What is a vector space?If V is a set of elements on which twooperations called vector additionand

vector multiplicationare defined then V isa vector space if the eight axioms onpage 10 are satisfied.

A vector space is a set of elements



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Example 1Examples of vector spaces

The set of real numbers, R

The set of ordered pairs, R

The set of triples, R

The set of complex numbers, The set of all m n matrices

The set of all polynomials of degree n


n

C


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Linear dependence

Linear combination- scalarmultiplication and vector addition.

The set of vectors is linearlydependent.


1 2c k a k b

, ,a b c


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Linear independenceConsider the set of vectors

The set is linearly independent if and

only if


1 2; ;... nx x x

1 21 2 ... 0nnk x k x k x

1 2 ... 0nk k k


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Example 2aShow that the unit vectors and in

two dimensions is linearly independent.

Note: Geometrically and areperpendicular and thus not multiples ofone another


i j

i j


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Example 2b) Are and

linearly independent?

c) Show that ,

and are linearlydependent.


a 2; 6 b 1; 3

a 1;1;1 2; 1; 4b

5;2;7c


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Basis and dimensionBasis: consider the set

of vectors, denoted by

in a vector space V. If this set is linearlyindependent in V such that

Then B is called a basis for V.


1 2B ; ;...; nx x x

1 2S ; ;...; nx x x

1 2S ; ;...; V

nx x x


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Basis and dimension Every vector in V can be written as a

unique linear combination

Any other basis of V must also containnlinearly independent vectors

The vector space has dimension n

(the number of vectors in a basis)


y

1 21 2y ... nnk x k x k x


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Example 3aIs the following set a basis for a threedimensional Euclidean space? Motivate.


1 1 3

0 , 2 , 2

1 3 5


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Example 3bIs the following set a basis for a threedimensional Euclidean space? Motivate.


1 1 1

0 , 2 , 2

0 0 3


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Next Week Revise eigenvalues and eigenvectors

(Maths III)

Note available on myTUTor

Class test 1a


lesson 2 vector spaces

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