Linear combination and span

Given vectors \(v_1,\dots,v_k\) and scalars \(c_1,\dots,c_k\), the vector \(c_1v_1+\cdots+c_kv_k\) is a linear combination; the scalars are called coefficients. For example, \(2(1,2)+3(0,-1)=(2,1)\). Asking whether a vector \(b\) is a linear combination of others is asking whether a certain system of equations has a solution, and it is solved by Gaussian elimination.

The span of \(v_1,\dots,v_k\) gathers all their linear combinations. It always contains \(0\), each \(v_i\), and any sum or multiple of its elements. Geometrically, the span of a nonzero vector is a line through the origin; that of two vectors in \(\mathbb{R}^3\) that are not multiples of each other is a plane through the origin.

Not every vector adds something new: since \((2,4)=2(1,2)\), the span of \((1,2)\) and \((2,4)\) is the same line as that of \((1,2)\). On the other hand, \((1,1)\) and \((1,2)\) generate all of \(\mathbb{R}^2\). Whether some vectors generate a given space and whether some are redundant are the two questions that lead to linear independence and to bases.

A common mistake is thinking that coefficients must be positive or integer; they may be any real number. Another is forgetting that in a general space (matrices, polynomials, functions) combinations follow the same rules: for instance, \(x^2\) and \(1\) generate the polynomials \(ax^2+b\).