Answer to Question #207337 in Geometry for Asma M.Hussain

Question

Linear combination and linear dependent set versus Affine combinations and Affine dependent sets

Explanation with detail .theorem.

Answer 1

 linear combination refers to an expression that is constructed from a set of values by multiplying a constant by each value and then adding the results for example a linear combination of x and y can be any expression with the form ax + by, where a and b are constants

A linear dependent set is a set of vectors that have nontrivial linear combination that equals the zero vector.

an affine combination lets say of vectors x1, ..., xn is a vector called a linear combination of x1, ..., xn, whereby the coefficients sum is 1, therefore, the vectors represent elements of a given vector space V over a field K, and the coefficients are scalars in K.

An affine dependent set is nontrivial if there is some i with αi = 0. ... An affine map f : d → k can be represented as a linear map combination and a translation. Therefore, the coordinates, f can be written as f(x) = Ax + b, where A is a real (k × d)-matrix and b ∈ k.