Vector Spaces Lect

8/8/2019 Vector Spaces Lect

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Vector spaces


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Numbers (Reals)Real numbers, , are the set of numbers that we

express in decimal notation, possibly with infinite,

non-repeating, precision.


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Numbers (Reals)Example: T=3.141592653589793238462643383279502884197

Completeness: If a sequence of real numbers getsprogressively tighter then it must converge to areal number.

Size: The size of a real numbera is the squareroot of its square norm:

2aa !


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Numbers (Complexes)

Complex numbers, , are the set of numbers that we

express as a+ib, where a,b and i= .

Example: eiU=cosU+isinU

1


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Numbers (Complexes)

Let p(x)=xn+an-1xn-1++a1x

1+a0 be a polynomial

with ai.

Algebraic Closure:

p(x) must have a root, x0 in :

p(x0)=0.


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Numbers (Complexes)Conjugate: The conjugate of a complex numbera+ib

is:

Size: The size of a real numbera+ib is the square

root of its square norm:

ibaiba !

22)()( baibaibaiba !!


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GroupsA group G is a set with a composition rule + thattakes two elements of the set and returns anotherelement, satisfying:

Asscociativity: (a+b)+c=a+(b+c) for all a,b,cG. Identity: There exists an identity element 0G such that

0+a=a+0=a for all aG.

Inverse: For every aG there exists an element -aGsuch that a+(-a)=0.

If the group satisfies a+b=b+a for all a,bG, thenthe group is called commutative.


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GroupsExamples:

The integers, under addition, are a commutative group.

The positive real numbers, under multiplication, are a

commutative group. The set of complex numbers without 0, under

multiplication, are a commutative group.

Real/complex invertible matrices, under multiplicationare a non-commutative group.

The rotation matrices, under multiplication, are a non-commutative group. (Except in 2D when they arecommutative)


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(Real) Vector Spaces

A real vector space is a set of objects/vectors that can be

added together and scaled by real numbers.Formally:

A real vector space Vis a commutative group with a scaling operator:

(a,v)av,

a, u,v,wV, such that:

1.1v=v for all vV.

2.a(v+w)=av+aw for all a, v,wV.

3.(a+b)v=av+bv for all a,b, vV.4.(ab)v=a(bv) for all a,b, vV.

5.v+w=w+v for all v,wV.

6.u+(v+w)=(u+v)+w for all, u, v,wV.


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(Real) Vector SpacesExamples:

The set of n-dimensional arrays with real coefficients is a

vector space.

The set of mxn matrices with real entries is a vector space.

The sets of real-valued functions defined in 1D, 2D,

3D, are all vector spaces.

The sets of real-valued functions defined on the circle,

disk, sphere, ball, are all vector spaces.

Etc.


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(Complex) Vector Spaces

A complex vector space is a set of objects that can beadded together and scaled by complex numbers.

Formally:

A complex vector space V is a commutative group with a scaling

operator:

(a,v)av,

a, vV, such that:

1.1v=

v for all vV.2.a(v+w)=av+aw for all a, v,wV.

3.(a+b)v=av+bv for all a,b, vV.

4.(ab)v=a(bv) for all a,b, vV.


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(Complex) Vector SpacesExamples:

The set of n-dimensional arrays with complex coefficientsis a vector space.

The set of mxn matrices with complex entries is a vectorspace.

The sets of complex-valued functions defined in 1D, 2D,3D, are all vector spaces.

The sets of complex-valued functions defined on the

circle, disk, sphere, ball, are all vector spaces. Etc.


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(Real) Inner Product SpacesA real inner product space is a real vector space V

with a mapping V,V that takes a pair of vectors

and returns a real number, satisfying:

u,v+w=u,v+ u,w for all u,v,wV.

u,v=u,v for all u,vVand all .

u,v=v,u for all u,vV.

v,vu0 for all vV, and v,v=0 if and only if v=0.


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(Real) Inner Product SpacesExamples:

The space ofn-dimensional arrays with realcoefficients is an inner product space.If v=(v1,,vn) and w=(w1,,wn) then:

v,w=v1w1++vnwn IfMis a symmetric matrix (M=Mt) whose eigen-

values are all positive, then the space ofn-dimensional

arrays with real coefficients is an inner product space.If v=(v1,,vn) and w=(w1,,wn) then:

v,wM=vMwt


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(Real) Inner Product Spaces

Examples:

The space ofmxn matrices with real coefficients is an

inner product space.


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(Real) Inner Product Spaces

Examples:

The spaces of real-valued functions defined in 1D,

2D, 3D, are real inner product space.Iffand gare two functions in 1D, then:

The spaces of real-valued functions defined on the

circle, disk, sphere, ball, are real inner productspaces.

Iffand gare two functions defined on the circle, then:

g

g! dxxgxfgf )()(,

!T

UUU2

0)()(, dgfgf


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(Complex) Inner Product Spaces

A complex inner product space is a complex vector

space Vwith a mapping V,V that takes a pair of

vectors and returns a complex number, satisfying:u,v+w=u,v+ u,w for all u,v,wV.

u,v=u,v for all u,vVand all .

for all u,vV.

v,vu0 for all vV, and v,v=0 if and only if v=0.

uv,vu, !


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(Complex) Inner Product SpacesExamples:

The space ofn-dimensional arrays with complexcoefficients is an inner product space.

If v=(v1,,vn) and w=(w1,,wn) then:

IfMis a conjugate symmetric matrix ( ) whoseeigen-values are all positive, then the space ofn-dimensional arrays with complex coefficients is an

inner product space.If v=(v1,,vn) and w=(w1,,wn) then:

v,wM=vMwt

nn11 wv...wvwv, !

tMM !


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Examples:

The space ofmxn matrices with real coefficients is an

inner product space.


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Examples:

The spaces of complex-valued functions defined in

1D, 2D, 3D, are real inner product space.Iffand gare two functions in 1D, then:

The spaces of real-valued functions defined on the

circle, disk, sphere, ball, are real inner productspaces.

Iffand gare two functions defined on the circle, then:

g

g! dxxgxfgf )()(,

!T

UUU2

0)()(, dgfgf


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Inner Product Spaces

IfV1,V2V, then Vis the direct sum of subspaces V1,V2, written V=V1V2, if:

Every vectorvVcan be written uniquely as:

for some vectors v1V1 and v2V2.21 vvv !


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Inner Product Spaces

Example:

IfVis the vector space of 4-dimensional arrays, then

Vis the direct sum of the vector spaces V1,V2Vwhere:

V1=(x1,x2,0,0)

V2=(0,0,x3,x4)


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Sub

space

A subspace is a subset of a vector space whichis a vector space itself, e.g. the plane z=0 is a

subspace of R3 (It is essentially R2.).

A subspace S of a vector space V is

nonempty set of vectors in Such that:

1. If a, b S, then a+bS.2. If aS, c , a.c S.


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Linear system & subspaces

Null space: {(c,c,-c)}

!

0

0

0

31

32

01

vu

!

0

0

0

431

532

101

z

y

x

The set of solutions to Ax = 0 forms a subspacecalled the nullspace of A.

Null space: {(0,0)}


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6.837 Linear Algebra Review

Orthonormal Basis Basis: a space is totally defined by a set of

vectors any point is a linear combination

of the basis

Ortho-Normal: orthogonal + normal

Orthogonal: dot product is zero

Normal: magnitude is one

Example: X, Y, Z (but dont have to be!)


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Orthonormal Basis

0

0

0

!

!

!

zy

zx

yx? A

? A

? AT

T

T

z

y

x

100

010

001

!

!

!

X,Y, Z is an orthonormal basis. We can describe any 3D

point as a linear combination of these vectors.

How do we express any point as a combination of a new basis

U, V,N, given X,Y, Z?


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Orthonormal Basis

-

!

-

-

ncnbna

vcvbva

ucubua

nvu

nvu

nvu

c

b

a

333

222

111

00

00

00

(not an actual formula just a way of thinking about it)

To change a point from one coordinate system to

another, compute the dot product of each coordinate rowwith each of the basis vectors.

Vector Spaces Lect

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