Dimension Calculator Linear Algebra

Calculator

Matrix A

Use spaces or commas. Separate rows by new lines.

Matrix B Optional

Use this for span, sum, and intersection dimensions.

Calculation Mode

Basis Vector Layout

Decimal Precision

Zero Tolerance

Example Data Table

Matrix A	Meaning	Expected Result
1 2 1 4 2 4 0 8 0 1 3 2	Three rows and four columns	Rank 3, nullity 1, left nullity 0
1 0 0 1 1 1	Optional Matrix B as column basis	Span dimension 2

Formula Used

Rank: Number of pivot columns in the row reduced echelon form.

Nullity: Nullity(A) = number of columns − rank(A).

Row space dimension: dim(Row A) = rank(A).

Column space dimension: dim(Col A) = rank(A).

Left nullity: Left nullity(A) = number of rows − rank(A).

Subspace sum: dim(U + V) = rank([A B]) when basis vectors are columns.

Intersection: dim(U ∩ V) = dim(U) + dim(V) − dim(U + V).

How to Use This Calculator

Enter Matrix A first. Put each row on a new line. Separate entries with spaces or commas. Use Matrix B only when comparing two spans. Choose whether basis vectors are stored in rows or columns. Select the calculation mode. Press Calculate. The result appears above the form.

Use the CSV button for spreadsheet work. Use the PDF button for a printable report. Change tolerance only when very small values should be treated as zero.

Dimension Calculator Linear Algebra Guide

What Dimension Means

A dimension calculator helps you study a matrix as a map. The rows and columns show how vectors move. The most useful number is rank. Rank counts the pivot columns after reduction. It also gives the dimension of the column space. The same value gives the dimension of the row space. These two spaces can look different. Yet their dimensions are always equal.

Rank and Nullity

Nullity is another key result. It counts free variables in a homogeneous system. A large nullity means many input vectors go to zero. A zero nullity means the columns are independent. For a matrix with n columns, rank plus nullity equals n. This is the rank nullity theorem.

Reduced Form

The calculator reduces your matrix to row reduced echelon form. This form makes pivots easy to see. Pivot columns form a basis pattern for the column space. Nonzero rows form a basis pattern for the row space. Free columns describe the solution space. The tool also reports left nullity. This value is m minus rank for an m by n matrix.

Subspace Comparison

You can also compare two subspaces. Enter bases as columns in the first and second matrices. The tool finds the dimension of each span. It then joins the columns. The rank of the joined matrix gives the dimension of the sum. The intersection dimension follows from the formula dim U plus dim V minus dim U plus V.

Study Uses

Use this page for homework checks, lecture review, and quick planning. It supports decimals, fractions written as decimals, negative values, and rectangular matrices. Results are rounded for display only. The calculations use a small tolerance to avoid floating noise. Always check exact symbolic work when your course needs proof.

Why It Helps

Dimension answers explain structure. They do more than solve one system. They show independence, dependence, constraints, and freedom. A matrix with full column rank has independent columns. A matrix with full row rank reaches the whole output space. These facts help with transformations, least squares, data models, and many applied problems.

For best results, keep each row the same length. Put missing entries as zero. Review pivot positions before using exported files. This habit makes errors easier to catch and explain during exams and regular practice.

FAQs

What does dimension mean in linear algebra?

Dimension is the number of independent directions needed to describe a vector space. For a matrix, it is often found through rank, nullity, row space, and column space.

What is rank?

Rank is the number of pivot columns after row reduction. It equals the dimension of the row space and the column space.

What is nullity?

Nullity is the number of free variables in Ax = 0. It tells how many independent solutions exist in the null space.

Can this calculator use rectangular matrices?

Yes. The calculator works with square and rectangular matrices. Determinant is shown only when the matrix is square.

How should I enter matrix values?

Write each row on a new line. Separate numbers with spaces or commas. Fractions like 1/2 are also accepted.

What does left nullity mean?

Left nullity is the number of rows minus rank. It is the dimension of the left null space of the matrix.

How do I compare two subspaces?

Enter one basis in Matrix A and another in Matrix B. Choose whether vectors are stored as rows or columns, then calculate.

Why use zero tolerance?

Zero tolerance treats very small numbers as zero. It helps reduce floating point noise during row reduction.