Column Space Calculator

Enter the six entries of a 3×2 matrix A to find the rank, a basis, and the geometric shape (point, line, or plane) of its column space Col(A) in ℝ³.

Quick Facts

Definition
Col(A) = span of A's columns
The column space of an m×n matrix is a subspace of ℝᵐ spanned by its column vectors.
Rank
dim(Col A) = number of pivot columns
Found by row reduction to echelon form; rank can never exceed min(m, n) — here, min(3, 2) = 2.

Your Results

Calculated
Rank of the matrix
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dim(Col A) — maximum possible is 2
Column space shape
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Geometric description in ℝ³
Basis of the column space
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Linearly independent pivot columns
Normal / direction vector
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Defines the plane or line the columns span

Ready

Enter the six entries of a 3×2 matrix, then press Calculate.

How the Column Space works

For a matrix A, the column space Col(A) — also called the range of A — is the set of every vector you can build as a linear combination of A's columns. Equivalently, Col(A) is the set of all vectors b for which the system Ax = b has a solution. This calculator works with a 3×2 matrix, so A has two column vectors living in ℝ³, and Col(A) is a subspace of ℝ³.

Formula and method

Let the two columns be c₁ = (a₁₁, a₂₁, a₃₁) and c₂ = (a₁₂, a₂₂, a₃₂). The calculator determines linear independence by computing the cross product c₁ × c₂:

  • If both columns are the zero vector, rank = 0 and Col(A) is just the origin {0}.
  • If the cross product is the zero vector (the columns are parallel, or one of them is zero), rank = 1 and Col(A) is the line through the origin in the direction of whichever column is nonzero.
  • If the cross product is nonzero, the columns are linearly independent, rank = 2, and Col(A) is the plane through the origin with c₁ × c₂ as its normal vector — the plane's equation is (c₁ × c₂) · x = 0.

This mirrors the standard row-reduction method (reduce A to echelon form and keep the original columns that land under a pivot) but is solved directly for the 3×2 case using the cross product, which is defined precisely for two vectors in ℝ³.

Common sources of error

  • Mixing up rows and columns: a₁₂ is row 1 of column 2, not row 2 of column 1 — double-check which entry goes where before reading results.
  • Assuming near-parallel means parallel: columns that are close to, but not exactly, scalar multiples of each other are still linearly independent and give rank 2.
  • Confusing column space with row space: they always share the same dimension (the rank) but are generally different subspaces spanned by different vectors.

Checking your result

Verify a rank-2 result by confirming the normal vector is genuinely perpendicular to both columns: the dot product of the normal with c₁ and with c₂ should both come out to 0 (allowing for tiny floating-point rounding). For rank 1, confirm the two columns really are scalar multiples of one another. For rank 0, both columns should show as exactly (0, 0, 0).

Applications

Column space comes up whenever you need to know which right-hand sides b make Ax = b solvable — in circuit analysis, computer graphics transformations, and least-squares curve fitting, among others. Its dimension (the rank) also tells you how much independent information the columns actually carry, which is the basis for rank-based data compression and for diagnosing redundant variables in a linear model.

Frequently Asked Questions

What is the column space of a matrix?
The column space of an m×n matrix A, written Col(A), is the set of all linear combinations of A's columns — equivalently, every vector b for which Ax = b has a solution. It is a subspace of ℝᵐ (ℝ³ for a 3-row matrix), and its dimension equals the matrix's rank.
How do you find a basis for the column space?
Row-reduce the matrix to echelon form and note which columns contain a pivot. The original columns in those positions (not the reduced ones) form a basis for Col(A). For a 3×2 matrix, this is equivalent to checking whether the two columns are linearly independent: if their cross product is nonzero, both columns form the basis; if it is zero, the columns are parallel (or one is the zero vector) and at most one vector is needed.
What does the rank tell you about the column space geometrically?
For a 3×2 matrix the rank is 0, 1, or 2. Rank 0 means both columns are the zero vector, so Col(A) is just the origin. Rank 1 means the columns are parallel (or one is zero), so Col(A) is a line through the origin. Rank 2 means the columns are linearly independent, so Col(A) is a full plane through the origin in ℝ³.
Is the column space the same as the row space?
No. The row space is spanned by a matrix's rows instead of its columns, and for a non-square matrix it lives in a different ℝⁿ. The row space and column space always share the same dimension — the rank — even though they are generally different subspaces.

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