Linear combination · aiwiki.page

A linear combination is an expression formed by multiplying elements of a vector space by scalars and adding the results. It is a fundamental construction in linear algebra, connecting vector equations, spanning sets, coordinate systems, and linear transformations. The elements being combined need not be geometric arrows: they may also be polynomials, functions, or matrices, provided they belong to a common vector space. (mtaylor.web.unc.edu)

Definition and scope

Let $V$ be a vector space over a field $F$ . Given vectors $v_1,\ldots,v_n\in V$ and coefficients $a_1,\ldots,a_n\in F$ , their linear combination is

a_1v_1+\cdots+a_nv_n=\sum_{i=1}^{n}a_iv_i.

The coefficients are scalars belonging to the field over which the space is defined. Common choices are the real numbers and complex numbers. Closure under addition and scalar multiplication ensures that the resulting element is again in $V$ . No requirement generally forces the coefficients to be positive or to sum to one. (mtaylor.web.unc.edu)

The ordinary algebraic definition uses a finite sum. Even when a collection contains infinitely many vectors, an individual linear combination uses only finitely many of them. Infinite series require an additional notion of convergence and are not automatically algebraic linear combinations. Taking every coefficient to be zero gives the zero vector; the empty sum is also conventionally the zero vector. (mtaylor.web.unc.edu)

For a polynomial, an expression such as

p(t)=3-2t+5t^2

is a linear combination of $1,t,t^2$ . Likewise, for functions sharing a domain and scalar field, $af+bg$ denotes the function whose value at $t$ is $af(t)+bg(t)$ . These examples illustrate why the definition is broader than coordinate-vector arithmetic. (mtaylor.web.unc.edu)

Geometric interpretation and span

In real coordinate spaces, scalar multiplication changes a vector’s length and, for a negative scalar, reverses its direction. Vector addition combines the resulting displacements. For example,

2(1,0)-3(0,1)=(2,-3).

A single nonzero vector generates a line through the origin. Two nonparallel vectors generate a plane through the origin; two parallel vectors generate only a line. Thus the geometry depends on the directions available, not simply on the number of listed vectors. (textbooks.math.gatech.edu)

The linear span of a set $S$ , written $\operatorname{span}(S)$ , is the set of all finite linear combinations of elements of $S$ . It is the smallest linear subspace containing $S$ . A combination is one particular expression or resulting vector, whereas a span is the entire collection of obtainable vectors. The span of the empty set is $\{0\}$ . (interactivetextbooks.tudelft.nl)

Independence, bases, and coordinates

A family $v_1,\ldots,v_n$ has linear independence when

a_1v_1+\cdots+a_nv_n=0

implies $a_1=\cdots=a_n=0$ . Otherwise, the family is linearly dependent. In a dependent finite family, at least one vector can be expressed as a linear combination of the others, making that vector unnecessary for generating the span. (interactivetextbooks.tudelft.nl)

Independence determines uniqueness: every vector in the span of an independent family has exactly one coefficient tuple relative to that family. To see this, subtract two proposed representations; independence forces every difference between corresponding coefficients to vanish. For a dependent family, a nonzero relation can instead be added to an existing representation to produce different coefficients with the same result. (interactivetextbooks.tudelft.nl)

A basis is an independent spanning set. Its ordered elements therefore provide coordinates: each vector is represented uniquely by its coefficients. In a finite-dimensional space, the number of basis elements is its dimension. (textbooks.math.gatech.edu)

Matrix formulation and computation

For coordinate vectors $v_1,\ldots,v_n\in F^m$ , place them as columns of a matrix:

A=\begin{bmatrix}v_1&\cdots&v_n\end{bmatrix}.

Then

Ax=x_1v_1+\cdots+x_nv_n.

Determining whether a target vector $b$ is a combination of the columns is equivalent to solving the system of linear equations $Ax=b$ . (textbooks.math.gatech.edu)

Gaussian elimination tests consistency and finds coefficients when a solution exists. The rank measures the dimension of the column span. A consistent system has unique coefficients precisely when its columns are independent; dependent columns permit multiple coefficient vectors for the same target. If the system is inconsistent, the target lies outside the column span. (textbooks.math.gatech.edu)

Restricted combinations

Several related constructions impose additional conditions on real coefficients:

An affine combination requires $\sum_i a_i=1$ .
A conical combination requires $a_i\geq0$ .
A convex combination requires both conditions. (math.mit.edu)

For distinct points $u,v$ , affine combinations $(1-t)u+tv$ , with unrestricted real $t$ , describe the entire line through them. Convex combinations restrict $t$ to $[0,1]$ , describing the connecting segment. These constructions underlie affine spaces and convex sets; nonnegativity is an extra condition, not part of a general linear combination. (math.mit.edu)

Linear maps and statistical models

A linear map preserves linear combinations:

T\left(\sum_i a_iv_i\right)=\sum_i a_iT(v_i).

Consequently, its values on a basis determine its values throughout the space. This property explains the close relationship between linear combinations and matrix representations of transformations. (textbooks.math.gatech.edu)

In linear regression, the fitted predictor is a linear combination of selected feature columns, often including a constant column for the intercept. Ordinary least squares selects coefficients to minimize squared residuals. Polynomial terms and interactions may serve as features: “linear” here refers to the coefficients’ role, not necessarily to dependence on the original input variables. (online.stat.psu.edu)