Linear Combinations

Why

We want to build vectors out of other vectors using scalar multiplication and vector addition.

Definition

A linear combination from a vector space is an ordered pair: the first coordinate is a sequence of vectors and the second is a sequence of scalars. The result of a linear combination is the sum of the results of scaling each vector by the corresponding scalar in the sequence; itself a vector in the space.

A trivial linear combination is one whose sequence of scalars is zero at each coordinate. The result of any trivial linear combination is the zero vector. A nontrivial linear combination is one which is not trivial. In other words, to be nontrivial, there must exist at least one index of the scalar sequence whose corresponding value is nonzero.

We say that a given vector can be written as a linear combination of a sequence of vectors if there exists a sequence of scalars such that the result of the linear combination of that sequence of vectors and scalars is the given vector. In other words, a vector can be written as a linear combination of some other vectors if there exists scalars for those other vectors such that scaling them and adding the results gives the vector.

Notation

Let $(V, \F )$ be a vector space. Let $v = (v_1, \dots , v_n)$ be a sequence of vectors in $V$ and $a = (a_1, \dots , a_n)$ be a sequence of scalars in $\F $. Then $(v, a)$ is a linear combination and we can express its result by

\[ a_1v_1 + a_2v_2 + \cdots + a_n v_n. \]

If $(v,a)$ is trivial, then $a_i = 0$ for $i \in \upto{n}$ and the result of $(v, a)$ is $0$ (the zero vector). Otherwise, there exists $i \in \upto{n}$ such that $a_i \neq 0$; of course, the result of $(v, a)$ may still be $0$. A vector $u$ can be written as a linear combination of the vectors $v_1, v_2, \dots , v_n$ if there exists scalars $a_1, a_2, \dots , a_n$ so that

\[ u = a_1v_1 + a_2v_2 + \cdots + a_nv_n. \]

We often number (see \sheetref{lists}{Lists}) a set of finite vectors. In this case, we say “Let $\set{u_1, \dots , u_n}$ be a (finite) set of vectors.”