Vectors, Linear Combinations and Span
A linear combination mixes vectors with scalars. The span is every vector you can reach that way, and independence tells you whether any vector is redundant.
Linear combinations
A linear combination of vectors v and w is any a·v + b·w. To reach (5, 2) using (1, 0) and (1, 1), solve a(1, 0) + b(1, 1) = (a + b, b) = (5, 2). The second entry gives b = 2, then a = 3. In the explorer, dragging the sliders for a and b moves the combined vector across the plane.
Span
The span of a set of vectors is the set of all their linear combinations. Two vectors that point in different directions span the whole plane ℝ². Two vectors on the same line, such as (1, 2) and (2, 4), only span that line.
Independence
Vectors are linearly independent when none of them is a combination of the others. For two vectors in ℝ², compute the determinant: for (1, 2) and (2, 4) it is 1·4 − 2·2 = 0, so they are dependent. A nonzero determinant means independent, and then they form a basis of the plane.
Common mistake
Dependent vectors are not "bad". They are simply redundant, and the dimension of the span is the number of independent vectors, which is also the rank of the matrix built from them.
Basis and dimension
A basis is an independent set that spans a space, and the number of vectors in it is the dimension. In ℝ³ you need three independent vectors to span everything. Two span a plane through the origin, and every span contains the zero vector.
Quick check
Is (4, 6) in the span of (2, 3)? Yes, it equals 2·(2, 3). Is (4, 7)? No: no single multiple of (2, 3) gives it, and one vector only spans a line. To test membership in general, put the vectors in the columns of a matrix and see whether Ax = b has a solution.