How this instrument works
A matrix's column space is every vector reachable by combining its columns. For a genuinely independent (full-rank) 2×2 matrix, that column space is already the entire plane — nothing further to reduce. This page instead covers the more interesting rank-deficient case, where the two columns are PROPORTIONAL (pointing the same direction, just scaled differently), collapsing the entire column space down to a single line.
Finding that line's simplest direction reuses a familiar reduction: dividing the first column's own two entries by their greatest common divisor, the exact same operation this site's Lowest Term calculator applies to fractions. What emerges is the smallest whole-number vector pointing along the shared direction every combination of the matrix's columns is confined to.
This page requires the matrix's columns to actually be proportional (checked via b×c=a×d, the condition for a zero determinant) — a full-rank matrix's column space needs no such reduction at all, since it already reaches every point in the plane.
- Enter the matrix's four entries into the a, b, c, and d fields.
- Confirm the columns are proportional — b×c should equal a×d.
- Read Column space basis direction: x and y — the simplest whole-number vector spanning that single line.
- Divide the first column's own two entries by the reported basis direction to confirm the same reduction factor applies to both.
Worked example — [[1,2],[2,4]]
For [[1,2],[2,4]]: the two columns, (1,2) and (2,4), point the identical direction, just scaled by a factor of 2 — so every combination of them still only ever reaches points along that one line. Column 1, (1,2), is already in lowest terms (its GCD is 1), so the basis direction is (1,2) itself.
For [[4,1],[8,2]]: column 1, (4,8), reduces by its GCD of 4 down to (1,2) — the identical spanning line as the previous example, just scaled up in the original matrix. For [[3,6],[0,0]]: column 1, (3,0), reduces by its GCD of 3 to (1,0) — a purely horizontal column space, since the entire second row is 0.
Questions
What is a matrix's column space?
The set of every vector reachable by combining a matrix's own columns — for a full-rank 2×2 matrix, that's the entire plane; for a rank-deficient matrix with proportional columns, it collapses down to a single line.
Why does this page only cover rank-deficient matrices?
A full-rank matrix's column space is already the whole plane, with no single spanning direction to reduce to — the reduction this page performs is only meaningful once the columns collapse down to one shared line.
How do you know if a matrix's columns are proportional?
Check whether b×c equals a×d — that's exactly the condition for the matrix's determinant to be zero, which is the same as the two columns pointing along a shared direction.
How is this related to the Lowest Term calculator on this site?
Both reduce a pair of whole numbers by dividing out their greatest common divisor — that page applies it to a fraction's numerator and denominator, while this page applies the identical reduction to a column vector's two entries.
What does the basis direction actually represent?
The smallest whole-number vector pointing along the single line every combination of the matrix's columns is confined to — every other vector in the column space is just a scalar multiple of this one direction.