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One can look at the operation of a matrix times a vector as changing the basis set for the vector or as changing the vector with the same basis description. Many signal and systems problem can be posed in this form.

A matrix times a vector

In this chapter we consider the first problem posed in the introduction

A x = b

where the matrix A and vector x are given and we want to interpret and give structure to the calculation of the vector b . Equation [link] has a variety of special cases. The matrix A may be square or may be rectangular. It may have full column or row rank or it may not. It may be symmetric or orthogonalor non-singular or many other characteristics which would be interesting properties as an operator. If we view the vectors as signals and thematrix as an operator or processor, there are two interesting interpretations.

  • The operation [link] is a change of basis or coordinates for a fixed signal. The signal stays the same, the basis (or frame) changes.
  • The operation [link] alters the characteristics of the signal (processes it) but within a fixed basis system. The basis stays the same, the signalchanges.

An example of the first would be the discrete Fourier transform (DFT) where one calculates frequency components of a signal which arecoordinates in a frequency space for a given signal. The definition of the DFT from [link] can be written as a matrix-vector operation by c = Wx which, for w = e - j 2 π / N and N = 4 , is

c 0 c 1 c 2 c 3 = w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 3 w 0 w 2 w 4 w 6 w 0 w 3 w 6 w 9 x 0 x 1 x 2 x 3

An example of the second might be convolution where you are processing or filtering asignal and staying in the same space or coordinate system.

y 0 y 1 y 2 = h 0 0 0 0 h 1 h 0 0 h 2 h 1 h 0 x 0 x 1 x 2 .

A particularly powerful sequence of operations is to first change the basis for a signal, then process the signal in this new basis, and finally returnto the original basis. For example, the discrete Fourier transform (DFT) of a signal is taken followed by setting some of the Fourier coefficients tozero followed by taking the inverse DFT.

Another application of [link] is made in linear regression where the input signals are rows of A and the unknown weights of the hypothesisare in x and the outputs are the elements of b .

Change of basis

Consider the two views:

  1. The operation given in [link] can be viewed as x being a set of weights so that b is a weighted sum of the columns of A . In other words, b will lie in the space spanned by the columns of A at a location determined by x . This view is a composition of a signal from a set of weights as in [link] and [link] below. If the vector a i is the i t h column of A , it is illustrated by
    A x = x 1 a 1 + x 2 a 2 + x 3 a 3 = b .
  2. An alternative view has x being a signal vector and with b being a vector whose entries are inner products of x and the rows of A . In other words, the elements of b are the projection coefficients of x onto the coordinates given by the rows of A . The multiplication of a signal by this operator decomposes the signal and gives thecoefficients of the decomposition. If a ¯ j is the j t h row of A we have:
    b 1 = a ¯ 1 x b 2 = a ¯ 2 x e t c .
    Regression can be posed from this view with the input signal being the rows of A .

Questions & Answers

an increase in demand (while supply remains constant) what will happen to deh graph?
Thabiso Reply
what is going to happen to the graph if there is an increase in demand, While supply remains constant .
What will happen to the graph if there is an increase in demand While supply remains constant?
price will increase high than automatically demand will decrease
equilibrium ?
what is demand
Sarkwah Reply
demand is the willingness to buy a commodity backed by the ability to pay.
demand is mere desire on commodity with ability to back up with purchasing power
Equilibrium is when there's an equality between quantity demanded and quantity supplied
Victory Reply
Again the consumer will be in equilibrium if the price of the commodity is equal to Marginal utility of that product
wat is the law of supply
Agnes Reply
It's what* -The law of supply states that price and supply is relative. As all factors are equal, if price increases then quantity of supply there for increases.
the law of suppy state that when prise is high, more commodity with be supply and when p is low less of the same commodity will be supply.
It states that, "other things being equal, move supplied at a higher price than at a lower price ".
it's state that the increased in prices will lead to decreased in supply
what is the theory of supply and the determinants of demand
And please what is change in quantity supplied?
guys why are you so quiet
A woman has a television set which cost her $800 two years ago. A new set would cost her $1000 and she could sell her television set for $450. What is the opportunity Cost of keeping the old TV?
Murewah Reply
principle of effective demand?
Abubakar Reply
the is the situation in which the need of individuals exceed the available resource. increase in population rate and wrong decision making
esther Reply
what is the different between wants and demand?
wants are what people desire to have but they can live without them and demand is a thing that is most wanted
what are the demand pull inflation
the higher the aggregate level of activity, the larger the proportion of areas and industries which experience excess demand for goods and labour of various sorts , and the more powerful is demand-inflationary pressure . Demand inflation is contrasted with cost inflation , in which price and wage
increases are transmitted from one sector to another. These should be regarded as different aspects of an overal inflation starts , cost inflation explains why inflation once begun is so difficult to stop.
what is the important difference between positive and normative economics
positive economics is the study of how an economy works in practice, as opposed to the theoretical study of how it should run in theory and normative economics is the party of economics that is concerned with how the economy ought to be run.
positive economic deal with fact and also talks about how the economy actually is like while normative economic deal with value judgement and talks about how the economy ought to be like
What is the difference between opportunity cost and choice
opportunity cost are also known as forgun alternative why choice is to select one among alternative
importance of economic
Zakaria Reply
satisfaction of human wants
economics is about to economise . discuss
Angel Reply
Underlines the efficiency aspect. Economise towards what: Economise factors to reach equal distribution of Material wealth or Just to operate optimally to Service demand, i. e. Run markets efficiently?
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abba Reply
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Ibrahim Reply
different btn import and export
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Gbenga Reply
hw can we solve problem of scarcity
scarcity is not necessarily a problem but a constant condition of the world. there are not enough resources to satisfy the unlimited wants.
wee need to be cooperative
by unlimited resourses and abundant want
What is the economic problem
And what is demand pull inflation
why do compute GDP?
steven Reply
can anyone shortly determine the word inflation.
Ibrahim Reply
Continous increase in the general level of prices or in the cost of living.
persistent increased in general price level
all correct...
rise in price.
deserving of money
A persistent tendency for nominal prices to increase
What is the problem of economic problem
the father of economics
Reuben Reply
Adem smith
Adem smith
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the father of political of economic and capitalism in his book and inquary in to the wealth of the nation.
Adam Smith his the father of economic
difference between injection and leakage
what is monopoly
Monopoly is a market structure where there is one firm who dominate the industry
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you are welcome
monopoly is the one characterized by a mkt power in which a firm is a price maker
Some member just ask questions but not answering so y this happen
Monopoly is a market where only one seller exists. No competition
how long does the patent right prevail the monopoly
no attempt
what is state farming
anybody to attempt
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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