SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Vector space Hom(V,W)

Vector Space Hom(V,W). Let V be some abstract n-dimensional vector space over field F and let W be some abstract m-dimensional vector space over field F. Then every linear transformation from V into W can be represented by some mxn matrix over field F. Moreover, every mxn matrix over F represents some linear mapping from V into W. Let A be the set of all mxn matrices over field F. The set A then represents the set of all linear transformations from V into W. This set of all linear mappings from V into W is itself a vector space over F. The vector space consisting of all linear mappings from V into W is denoted by Hom(V,W) and has a dimension of mn i.e. it has mn linearly independent basis vectors.

Basis for Hom(V,W). Let

{v_{1}, v_{2}, ... ,v_{n }} are a set of basis vectors for V

{w_{1}, w_{2}, ... ,w_{m }} are a set of basis vectors for W.

A set of basis vectors for the vector space Hom(V,W) is given by the set of mn functions {F_{ij}:
i=1,n; j=1,m} where

F_{ij} maps v_{i } into w_{j }and all other v's into 0

Example. Let V be Euclidean 3-space and W be Euclidean 2-space. Let the set {x,y,z} be three basis vectors for vector space V and {x',y'} be two basis vectors for W. Then a basis for Hom(V,W) would be the following six functions:

Function F_{11}:

x x'

y 0

z 0

Function F_{12}:

x y'

y 0

z 0

Function F_{21}:

x 0

y x'

z 0

Function F_{22}:

x 0

y y'

z 0

Function F_{31}:

x 0

y 0

z x'

Function F_{32}:

x 0

y 0

z y'

Basis of the vector space A of all mxn matrices over a field F. A basis for the vector space A of all mxn matrices over a field F is given by the set of nm mxn matrices

{E_{ij}: i=1,m; j=1,n}

where E_{ij} has a 1 in the i-th row and j-th column, all other entries being zero.

Question. Are F_{ij} and E_{ij} related? If so, how are they related? Hom(V,W) corresponds
directly to the set A of all mxn matrices. The set of mn mxn matrices {E_{ij}: i=1,m; j=1,n}
represent a basis for A and so they must also represent a basis for Hom(V,W). But the set {F_{ij}:
i=1,n; j=1,m} also represent a basis for Hom(V,W). Is it the same basis? Do F_{ij} in any cases
correspond directly to E_{ij}? Each of the functions F_{ij} represent a linear mapping from space V
into space W and correspond to some member of the set Hom(V,W) of all such mappings. Thus
each F_{ij} must correspond to some mxn matrix.

Answer. Let the set of n elementary n-vectors

e_{1} = (1, 0, 0,..., 0)

e_{2} = (0, 1, 0,..., 0)

...................

e_{n} = (0, 0, 0,..., 1)

constitute a basis for V. Let the set of m elementary m-vectors

g_{1} = (1, 0, 0,..., 0)

g_{2} = (0, 1, 0,..., 0)

...................

g_{m} = (0, 0, 0,..., 1)

constitute a basis for W. Consider the effect of the linear mapping of a basis vector e_{i} into a
vector w by the matrix E_{ij}

w = E_{ij} e_{i}

Here E_{ji} selects the i-th element of elementary vector e_{i} and maps it into the j-th position in vector
w, putting zeros elsewhere. Thus it maps the i-th elementary vector of V into the j-th elementary
vector of W. Hence it does the same thing as F_{ij} . In this case F_{ij} corresponds to the matrix E_{ji}.

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]