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Column Space Calculator (basis of Col(A), rank, pivot columns)

A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.

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Open Column Space Calculator (basis of Col(A), rank, pivot columns) and you get an answer straight away, with no account to create. Its place is under Matrices & vectors; Null Space Calculator (basis of Nul(A), nullity, parametric solution) and Matrix Rank Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Column Space Calculator (basis of Col(A), rank, pivot columns)?

A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is Column Space Calculator (basis of Col(A), rank, pivot columns) different from Null Space Calculator (basis of Nul(A), nullity, parametric solution)?

They sit next to each other but answer different questions: Null Space Calculator (basis of Nul(A), nullity, parametric solution) is the one to open when you need it to solve Ax = 0 for any matrix up to 8×8: a basis of the kernel in exact fractions and in cleared integers, the free variables, the parametric solution, the rank–nullity check and A·v = 0 verified exactly. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Matrix Rank Calculator is the closest one after this: The rank of a matrix by row reduction — the number of linearly independent rows.

What else is worth having open alongside it?

Gram-Schmidt calculator (orthonormal basis) and Spearman Rank Correlation Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

All guides →
ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.