Visualize the span of vectors in 2D and 3D. Explore linear independence, basis, dimension, and column/null spaces.
The span of a set of vectors is everything you can reach by scaling and adding them, and this tool draws exactly that shape. Enter vectors in 2D or 3D and it renders the flat through the origin that holds every linear combination - a line, a plane, or the whole space - along with the dimension, a linear-independence verdict, and whether the vectors form a basis. The common surprise: adding vectors does not automatically grow the span. A third vector lying in the plane of the first two changes nothing; only a genuinely new direction adds a dimension.
A concrete input and expected output from the current implementation.
Input
v1 = (1, 2) and v2 = (2, 4) in 2D
Expected output
Linearly dependent: v2 = 2v1. The span is the single line through the origin y = 2x, with dimension 1. The two vectors do not form a basis of the plane.
Every combination a(1,2) + b(2,4) equals (a + 2b)(1,2), so the span is all scalar multiples of (1,2) - the line y = 2x. Both vectors point along one direction, so they cover no more than a single vector would: the span has dimension 1, and since a basis of the plane needs two independent vectors, this set is not one.
Each entered vector becomes a column of a matrix, and the span is the set of all linear combinations of those columns. The tool counts how many columns are independent - the matrix rank - and draws the corresponding flat: a line through the origin for one, a plane for two, all of space for three in 3D. Adding a dependent vector changes none of this.
Vectors are independent when no nonzero combination of them equals zero, and dependent when one is a combination of the others. A set is a basis when its vectors are independent and span the whole space. For matrices it adds two subspaces: the column space, the span of the columns, and the null space, the vectors sent to zero. Rank-nullity links them: rank plus nullity equals the column count.
No. Three vectors span all of space exactly when they are linearly independent - none is a combination of the other two, equivalently the 3x3 determinant is nonzero. When the determinant is zero they lie in one plane through the origin, and the span is that plane or something smaller. Replace one vector with one pointing out of the plane and the span becomes all of space.
Yes, a single point is possible: the origin. If every vector you enter is the zero vector, the span shrinks to that one point, a subspace of dimension 0. More typically the span is a line (one independent vector), a plane (two), or all of space (three). The dimension always equals the number of independent vectors, never the number you typed.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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