AlgebraFree

Column Space Calculator

Use this free column space calculator to find the column space (range) of a 2x2 matrix. Determine whether the columns span all of R squared or just a line, get the rank, column independence status, and basis for the column space.

Enter Values

a (Row 1, Col 1)

Top-left entry of the matrix

b (Row 1, Col 2)

Top-right entry of the matrix

c (Row 2, Col 1)

Bottom-left entry of the matrix

d (Row 2, Col 2)

Bottom-right entry of the matrix

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

\text{Col}(A) = \text{span}(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n)

How it works: The column space is the set of all possible outputs Ax as x varies over all vectors. For a 2x2 matrix, check the determinant: if nonzero, the columns are independent and span all of R squared. If zero, the columns are dependent and the column space is a line through the origin.

Review and Methodology

Updated Mar 26, 2026

This calculator runs locally in your browser. Inputs are converted into the units required by the formula, and the result is paired with supporting references so you can verify the method before using it for planning or estimates.

Worked Example

A = [[1, 2], [2, 4]]

1Step 1: Column 1 = [1, 2], Column 2 = [2, 4]

2Step 2: Column 2 = 2 x Column 1 (dependent)

3Step 3: Determinant = (1)(4) - (2)(2) = 0 (confirms dependence)

4Step 4: Rank = 1

5Step 5: Column space basis = {[1, 2]}

Result: Col(A) = span{[1, 2]}, a line through the origin with slope 2.

What Is the Column Space of a Matrix?

The column space (also called the range or image) of a matrix A is the set of all possible output vectors Ax. It is spanned by the columns of the matrix. The column space tells you which vectors b make the system Ax = b solvable, which is one of the most fundamental questions in linear algebra.

If the column space is all of R squared (rank 2), the system Ax = b has a solution for every possible b
If the column space is a line (rank 1), only vectors b lying on that line can be reached
The dimension of the column space is the rank of the matrix
The column space and null space are complementary: rank + nullity = number of columns
In least squares regression, the predicted values are the projection of b onto the column space of the design matrix

The column space answers the question "what can this matrix produce?" For a 2x2 matrix with rank 2, the answer is any 2D vector. For rank 1, only vectors along a specific line. For the zero matrix (rank 0), only the zero vector. Understanding column space is essential for solving systems and understanding linear transformations.

You can also calculate changes using our Null Space Calculator, 2x2 Determinant Calculator, Matrix Rank Calculator or Linear Transformation Calculator.

Frequently Asked Questions

What does "full column space" mean for a 2x2 matrix?

It means the columns span all of R squared (the entire 2D plane). This happens when the determinant is nonzero, meaning the columns are linearly independent. The system Ax = b has a solution for every vector b, and the rank equals 2.

What if the determinant is zero?

Zero determinant means the columns are linearly dependent. One column is a scalar multiple of the other. The column space collapses from the full plane to a line through the origin, and the rank drops to 1 (or 0 for the zero matrix). The system Ax = b only has solutions when b lies on that line.

How is column space related to the range of a transformation?

They are the same thing. The column space of A equals the range (image) of the linear transformation T(x) = Ax. It is the set of all vectors that can be "reached" by applying the transformation to some input.

What is the difference between column space and row space?

Column space is spanned by the columns of A; row space is spanned by the rows. They always have the same dimension (the rank), but they live in potentially different spaces. For a 2x2 matrix, both are subspaces of R squared.

How do I check if a vector b is in the column space?

Try to solve Ax = b. If a solution exists, b is in the column space. For a 2x2 matrix with rank 1, check if b is a scalar multiple of the basis column vector. If it is, b is in the column space.

How is column space used in regression?

In least squares regression, the prediction y-hat is the projection of the observation vector y onto the column space of the design matrix X. The residual (y - y-hat) is orthogonal to the column space. This geometric view explains why the formula involves X transpose X.

How can I put this Column Space Calculator on my blog or website?

Yes, the Column Space Calculator is fully embeddable. Tap "Embed" above to configure appearance and copy the code. It is free to use, works on any platform (HTML, WordPress, CMS), and adjusts to any screen size automatically. Visit calculory.com/services/embed-calculators for the complete guide.

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