How this calculator works
The column space is the set of all linear combinations of the columns, which is also the set of vectors b for which Ax = b has a solution. Row reduction identifies pivot columns: the matching columns of the original matrix, not of the reduced matrix, form a basis. The dimension of the column space is the rank. It is at most the smaller of the row and column counts.
Formula
Col(A) = {Ax : x in R^n} = span of the pivot columns of A. dim Col(A) = rank(A).
Worked example
The matrix with rows 1, 2, 3 and 2, 4, 6 has one pivot column, column 1. A basis is the single vector 1, 2, so the column space is a line in the plane. The matrix with rows 1, 2 and 3, 4 has two pivot columns and its columns span the whole plane.
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Frequently asked questions
Which columns form the basis?
The pivot columns of the original matrix. Do not use the columns of the reduced matrix, because row operations change the column space.
How is this related to the null space?
Rank-nullity connects them: the dimension of the column space plus the nullity equals the number of columns.
Can the column space be all of R^n?
Yes, when the columns span n-dimensional space. That requires at least n columns and a rank of n.