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subject: Linear Combination [print this page]

In linear function, the linear models are Maximized or minimized which is used to set a linear constraints.Objective function variable,Set of constraints for linear,set of decision variables for linear are the important components of linear models.Most of the real world problem are leads to linear models components. Most of the real world problem in linear algebra can be approximated by using linear models.

Definition to Linear

Line segment present in the linear algebra are having only one dimension. This line segment is characterized by using the composition of the dimension present in the vector and linear numbers. The linear segment present in the linear algebra should be narrow.The linear segment also be elongated by using the parallel margins.These paralel margins are present in a linear leaf.

Linear combination is the central concept in linear algebra. Linear combination is mainly used in three concepts. They are

Functions

Vectors

Polynomials

In this topic we will briefly discuss how the linear combination applied and used in functions and vectors and polynomials and their properties.

Linear Combination - Defined

Functions:

In functions the linear combination is represented as sum of real part and imaginary part terms. Here we have to discuss about complex numbers. Complex number is the sum of the real part and imaginary part. It is defined by f(t)=eit and g(t)= e-it.

Then the cos function is represented as cos ht = (1/2) (eit + e-it)

Then the sin function is represented as sin ht = i(e-it - eit )

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Some properties of functions:

The sum of two functions is also the function. That is expressed as f(x) + g(x) = ( f + g) x

The product of two functions is also the function. That is expressed as f(x) g(x) = (f g) x

Vectors:

In vectors the linear combination is represented as the sum of its ordered pairs. It takes the general form as,

(p1, p2, p3) = (p1, 0, 0) + (0, p2, 0) + (0, p3, 0).

=p1 (1, 0, 0) + p2 (0, 1, 0) + p3 (0, 0, 1)

=p1e1+p2e2+p3e3

Some properties of vectors:

The sum of two vectors is also a vector

The product of two vectors is also a vector

Polynomials:

The general form of the linear combination is a1 (p1) + a2 (p2) +a3 (p3) = given polynomial

Here a1, a2, a3 are the arbitrary terms. And p1, p2, p3 given polynomials terms. We have to multiply the arbitrary and polynomial terms and comparing the coefficient terms of x to the right hand side value. These give the arbitrary values of a1, a2 and a3.

Some properties of polynomials:

The addition of two polynomials is also the polynomial.

The product of two polynomials is also the polynomial.

by: nitinp

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