Study vector spaces with rank and nullity checks. Paste matrices and review basis insights daily. Export results quickly for classes, homework, notes, and revision.
| Example Matrix | Rank | Nullity | Row Space Dimension | Column Space Dimension | Kernel Dimension |
|---|---|---|---|---|---|
| [1 2 3 4], [2 4 6 8], [0 1 1 1] | 2 | 2 | 2 | 2 | 2 |
| [1 0 0], [0 1 0], [0 0 1] | 3 | 0 | 3 | 3 | 0 |
Rank: The dimension of the row space equals the dimension of the column space.
Nullity: Nullity = number of columns − rank.
Rank-Nullity Theorem: dim(domain) = rank + nullity.
Linear map view: For a matrix A of size m × n, A maps Rn to Rm.
Invertibility test: A square matrix is invertible exactly when rank = n and nullity = 0.
Enter the number of rows and columns first. Keep the matrix size consistent with your pasted data.
Paste the matrix into the text area. Use one line for each row.
Separate entries with spaces, commas, or semicolons. Integers and decimals both work.
Press the calculate button. The result will appear above this form.
Review the summary, pivot columns, and RREF matrix. Export the result as CSV or PDF when needed.
Dimension tells you how many independent directions a space has. This idea is central in linear algebra. It explains structure, freedom, and redundancy. A matrix can describe a transformation, a system, or a data model. Dimension helps you read that matrix with more clarity. It also shows whether information is lost.
This calculator studies a matrix through row reduction. It finds rank first. Rank gives the dimension of the row space. It also gives the dimension of the column space. Then it finds nullity. Nullity measures the dimension of the kernel. Together, these values describe how the matrix behaves as a linear map.
The rank-nullity theorem connects the main results. If a matrix has n columns, then the domain dimension is n. That domain splits into image directions and kernel directions. Rank counts the image directions. Nullity counts the kernel directions. Their sum equals the domain dimension. This identity is useful in theory and in applications.
Reduced row echelon form makes the independent structure visible. Pivot columns show where independent information begins. Non-pivot columns depend on earlier pivot columns. Zero rows show loss of dimension. The calculator displays the reduced matrix so you can verify the result. This is helpful for homework, teaching, and checking manual work.
Dimension analysis appears in many settings. It helps when solving linear systems. It helps when checking if vectors span a space. It also helps in data science, graphics, control, and optimization. A full rank square matrix is invertible. A matrix with positive nullity has nontrivial kernel directions. Those facts guide many decisions.
Students can test conjectures fast. Teachers can build examples quickly. Analysts can inspect model matrices before deeper work. A small dimension change can alter solvability, stability, and interpretation in important ways often.
Start with the matrix size. Then read the rank. Next, compare nullity with the number of columns. Check the pivot column list. Review the invertibility line if the matrix is square. Finally, inspect the reduced matrix. These steps turn a list of values into a meaningful linear algebra interpretation.
Dimension is the number of independent vectors needed to describe a space. It tells you how many directions are available without redundancy.
Rank is a dimension. It equals the dimension of the row space and the column space of a matrix.
Nullity is the dimension of the null space. It counts how many free directions solve Ax = 0.
They are different spaces, but linear algebra proves they always have the same dimension. That common value is the rank.
A square matrix is invertible when its rank equals its order. In that case, nullity is zero and every column is a pivot column.
Yes. The calculator accepts integers, decimals, and negative numbers. It also accepts scientific notation because numeric parsing follows standard numeric rules.
Pivot columns identify independent columns. They help you choose a basis for the column space from the original matrix.
RREF makes the structure easy to verify. You can see pivots, zero rows, and dependent columns without doing extra manual reduction.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.