subject: Linear Combination [print this page] 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 )
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,
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.
Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is
a_1 v_1 + a_2 v_2 + a_3 v_3 + cdots + a_n v_n. ,
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, like in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace"; however one could also say "two different linear combinations can have the same value" in which case the expression must have been meant. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not give distinct linear combinations.
