VECTOR SPACE

APPLICATIONS

1. Space Flight and Control Systems

• Twelve stories high and weighing 7S tons, Columbia Rose

Majestically off the launching pad on a cool palm Sunday

morning in April 1981.

• A product of ten years intensive research and development, the

first U.S. Space shuttle was a triumph of control systems

engineering design, involving many branches of engineering-

aeronautical, chemical, electrical, hydraulic, and mechanical.

• The space shuttle's control systems are absolutely critical for

flight. Because the shuttle is an unstable airframe, it requires

constant computer monitoring during atmospheric flight. The

flight control system sends a stream of commands to

aerodynamic control surfaces and 44 small thruster jets.

• Figure I shows a typical closed loop feedback system that

controls the pitch of the shuttle during flight. (The pitch is the

elevation angle of the nose cone.)

• The junction symbols (⨂) show where signals from various

sensors are added to the computer signals flowing along the

top of the figure.

• Mathematically, the input and output signals to an

engineering system are functions. It is important in

applications that these functions can be added, as in fig. 1,

and multiplied by scalars.

• These two operations on functions have algebraic properties that are

completely analogous to the operations of adding vectors in 𝑅𝑛and

multiplying a vector by a scalar. For this reason, the set of all possible

inputs (functions) is called a vector space.

• The mathematical foundation for systems engineering rests on vector

spaces of functions , and the topic we are going to discuss extends the

theory of vectors in 𝑅𝑛 to include such functions.

• Later on, you will see how other vector spaces arise in engineering.

2. Control (or Guidance): Problems of control are associated

with dynamic systems evolving in time.

• Control or guidance usually refers to directed Influence on a

dynamic system to achieve desired performance.

• The system itself may be physical in nature, such as a rocket

heading for mars or a chemical plant processing acid, or it

may be operational such as a warehouse receiving and filling

orders.

• Often we seek feedback or so called closed-loop control in

which decisions of current control action are made

continuously in time based on recent observations of system

behaviour.

• Thus, one may imagine himself as a controller sitting at the

control panel watching meters and turning knobs or in a

warehouse ordering new stock based on inventory and

predicted demand.

• As an example of a control problem, we consider the launch

of a rocket to a fixed altitude h in given time T while

expending a minimum of fuel.

• For simplicity, we assume unit mass, a constant gravitational

force, and the absence of aerodynamic forces. The motion of

a rocket being propelled vertically is governed by the

equations

𝑦 = 𝑢 𝑡 − 𝑔

Where, y is the vertical height, u is the accelerating force, and

g is the gravitational force.

• The optimal control function u is the one which forces y(T) = h

while minimizing the fuel expenditure 0𝑇𝑢(𝑡) 𝑑𝑡.

• This problem might be formulated in a vector space consisting

of functions u defined on the interval [0, T].

• The solution to this problem, however, is that u(t) consists of an

impulse at t = 0 and, therefore, correct problem formulation

and selection of an appropriate vector space are themselves

interesting aspects of this example.

5. Allocation:

• In allocation problems there is typically a collection of

resources to be distributed in some optimal fashion.

• A typical problem of this type is that faced by a

manufacturer with an inventory of raw materials.

• He has certain processing equipment capable of producing n

different kinds of goods from the raw materials.

• His problem is to allocate the raw materials among the

possible products so as to maximize his profit.

• Assume that the selling price per unit of product j is pj,

j = 1,2, ... , n. If xj denotes the amount of product j that is to

be produced, bi the amount of raw material i on hand, and aij

the amount of material i in one unit of product j, the

manufacturer seeks to maximize his profit

𝑝1𝑥1 + 𝑝2𝑥2 +⋯+ 𝑝𝑛𝑥𝑛

Subject to the production constraints on the amount of raw materials

.

.

and

𝑥1 ≥ 0, 𝑥2 ≥ 0,… , 𝑥𝑛≥ 0.

𝑎11𝑥1 + 𝑎12𝑥2 +⋯+ 𝑎1𝑛𝑥𝑛 ≤ 𝑏1𝑎21𝑥1 + 𝑎22𝑥2 +⋯+ 𝑎2𝑛𝑥𝑛 ≤ 𝑏2

𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 +⋯+ 𝑎𝑚𝑛𝑥𝑛 ≤ 𝑏𝑚

• This type of problem, characterized by a linear objective function

subject to linear inequality constraints is linear programming problem

and is used to illustrate aspects of the general theory of optimization.

• We note that the problem can be regarded as formulated in ordinary

N-dimensional vector space. The vector x with components xi is the

unknown.

• The constraints define a region in the vector space in which the selected

vector must lie. The optimal vector is the one in that region which

maximizing the objective.

DRIVEN TO ABSTRACTION

• Generalization is necessary in linear algebra because studying 𝑅𝑛 ,

take us only so far. But as, many other sets of mathematical objects

such as functions, matrices, infinite sequences, etc., have properties

common with 𝑅𝑛, which suggest that we should generalize our

discussion of vectors to other sets of objects, which we call vector

space.

• By studying vector spaces whose objects share many of the same

properties of vectors in 𝑅𝑛, we reveal a more abstract theory with a

wider range of applications than we would obtain from a study of 𝑅𝑛

alone.

A LITTLE BIT OF HISTORY

• Vectors were first used about 1636 in 2D and 3D to describe geometrical

operations by René Descartes and Pierre de Fermat. In 1857 the notation of

vectors and matrices was unified by Arthur Cayley. Giuseppe Peano was the first

to give the modern definition of vector space in 1888, and Henri Lebesgue

(about 1900) applied this theory to describe functional spaces as vector spaces.

VECTOR SPACE

VECTOR SPACE

• A Vector Space is a non-empty set, V, of objects (called vectors) in which we

define two operations: the sum among vectors and the multiplication by a

scalar (an element of any field, R), and for u, v, w ∈ V and c, d ∈ R it is

verified that

1 u + v ∈ V 6 cu ∈ V

2 u + v = v + u 7 c (u + v) = c u + c v

3 (u + v) + w = u + (v + w) 8 (c + d) u = c u + d u

4 ∃0 ∈ V ∋ u + 0 = u 9 c(du) = (cd)u

5 u ∈ V ∃ 𝐰 ∈ V ∋ u + w = 0 10 1u = u

(we normally write w = −u)

EXAMPLES

Ex-1 Check whether the set V of all 2 x 2 matrices with real entries is Vector Space

or not if vector addition is defined to be matrix addition and vector scalar

multiplication is defined to be matrix scalar multiplication.

Let u, v, w ∈ V and c, d ∈ R

Axiom 1

Axiom 2

22222121

12121111

vuvu

vuvuvu

, 2221

1211

uu

uuu

2221

1211

vv

vvv

2221

1211

wand ww

ww

∴ u + v ∈ V

2221

1211

2221

1211

vv

vv

uu

uuvu

2221

1211

2221

1211

uu

uu

vv

vvuv

Axiom 3

Axiom 4

Axiom 5

2221

1211

22222121

12121111

)(ww

ww

vuvu

vuvuwvu

222222212121

121212111111

wvuwvu

wvuwvu

22222121

12121111

2221

1211

wvwv

wvwv

uu

uu

)( w vu

,00

00V

0 uu0

2221

1211

2221

1211

00

00

uu

uu

uu

uu

,2221

1211

Vuu

uu

u 0uu

00

00)(

2221

1211

2221

1211

uu

uu

uu

uu

Axiom 6

Axiom 7

Axiom 8

Vcucu

cucucuRc

2221

1211

,

22222121

12121111

22222121

12121111

)(cvcucvcu

cvcucvcu

vuvu

vuvucuc v

2221

1211

2221

1211

2221

1211

2221

1211

vv

vvc

uu

uuc

cvcv

cvcv

cucu

cucuvccu

2221

1211

)()(

)()()(,,

udcudc

udcudcudcRdc

ducududu

dudu

cucu

cucu

2221

1211

2221

1211

Axiom 9

Axiom 10

As all the propertice of Vector Space are satisfied, the set V of all 2 x 2 matrices with real entries is

Vector Space.

Example – 2

Is the set consisting of all m x n matrices together with standard matrix addition and scalar

multiplication vector Space?

Example – 3

Is the set consisting of all n x n singular matrices together with standard matrix addition and

scalar multiplication vector Space?

Example – 4 V be the set consisting of only second degree polynomials with standard addition

and scalar multiplication. Is V a vector space?

ucduu

uucd

dudu

duducduc )()(

2221

1211

2221

1211

uu

2221

1211

2221

121111

uu

uu

uu

uu