Span: everywhere a set of vectors can reach
Take some vectors and form every possible linear combination of them — every sum of scaled copies. The set of all those results is their span. It answers one question: starting from these vectors and allowed only to add and scale, where can you get to?
One non-zero vector spans a line through the origin. Two vectors pointing in genuinely different directions span the whole plane. But if the second vector is just a multiple of the first, it adds no new direction, and together they still span only a line. That distinction is the whole next idea.
Linear independence: is anyone redundant?
A set of vectors is linearly independent when no vector in it can be written as a combination of the others — every vector contributes a direction the rest cannot reach. If one of them can be built from the others, the set is dependent, and that vector is redundant.
The clean test: the vectors are independent exactly when the only way to combine them into the zero vector is to take all coefficients equal to zero. If some non-trivial combination collapses to zero, they are dependent.
Independence, determinant and invertibility are one idea
Put n vectors as the columns of an n×n matrix. They are independent ⇔ the determinant is non-zero ⇔ the matrix is invertible ⇔ the columns span the whole space. Dependent columns ⇔ determinant 0 ⇔ singular. It is the same fault line, seen from four directions.
Basis and dimension
A basis is the Goldilocks set: independent (nothing wasted) and spanning (enough to reach everything). It is the smallest set of vectors that still describes the entire space.
The number of vectors in a basis is the dimension of the space — and it is the same no matter which basis you choose. The plane has dimension 2, ordinary space dimension 3. Every vector in the space is a unique linear combination of a basis, which is exactly why coordinates work.
Testing it in practice
To check whether a handful of vectors are independent, make them the columns of a matrix and row-reduce. A pivot in every column means independent; a column with no pivot means that vector is a combination of the earlier ones. For the square case the shortcut is the determinant: non-zero means independent.
Are [1, 2] and [2, 4] independent? What about [1, 2] and [3, 1]?
- First pair: notice [2, 4] = 2·[1, 2].The second is a scalar multiple of the first, so it adds no new direction.
- det[[1, 2], [2, 4]] = (1)(4) − (2)(2) = 0 → dependent; together they span only a line.Zero determinant confirms the columns are dependent.
- Second pair: det[[1, 3], [2, 1]] = (1)(1) − (3)(2) = −5 ≠ 0 → independent.Non-zero determinant means the two vectors point in different directions.
[1,2] and [2,4] are dependent (they span a line); [1,2] and [3,1] are independent and form a basis for the plane.