Calculation Result
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Detailed Steps
Advanced Calculator
Enter a matrix, or use any two theorem values to find the missing value.
Example Data Table
Use these examples to compare matrix rank, nullity, and domain dimension.
| Example Matrix | Size | Domain Dimension | Rank | Nullity | Meaning |
|---|---|---|---|---|---|
[[1,2,3],[0,1,4]] |
2 × 3 | 3 | 2 | 1 | One free variable exists. |
[[1,0,0],[0,1,0],[0,0,1]] |
3 × 3 | 3 | 3 | 0 | The map is one-to-one. |
[[1,2,3,4],[2,4,6,8],[0,1,1,2]] |
3 × 4 | 4 | 2 | 2 | Two free variables exist. |
[[0,0],[0,0]] |
2 × 2 | 2 | 0 | 2 | Every input lies in the kernel. |
Formula Used
The calculator uses the rank-nullity theorem for a linear map
T: V → W. If the domain has dimension n, then:
rank(T) + nullity(T) = dim(V)
For a matrix A with m rows and n columns,
the domain dimension is n. The rank is the number of pivot columns.
The nullity is the number of free variables.
nullity(A) = n - rank(A)rank(A) = n - nullity(A)dim(domain) = rank(A) + nullity(A)left nullity = m - rank(A)
When matrix mode is selected, the script performs row reduction. It finds pivot columns, counts pivots, and applies the theorem to calculate nullity.
How to Use This Calculator
- Select whether you want to use a matrix or known theorem values.
- Enter the matrix rows and columns when matrix mode is active.
- Paste matrix entries with spaces, commas, tabs, or semicolons.
- For known values, enter any two among domain dimension, rank, and nullity.
- Choose decimal places and tolerance for row reduction.
- Press Calculate to show results below the header and above the form.
- Use CSV or PDF buttons to save your work.
Rank Nullity Theorem Guide
Why the theorem matters
The rank-nullity theorem connects two important ideas in linear algebra. Rank measures the output space reached by a linear map. Nullity measures the input directions that collapse to zero. Together, they account for every independent direction in the domain. This makes the theorem useful when checking matrix behavior. It also helps when studying systems of equations.
How the calculator helps
This calculator gives two working methods. You may enter a matrix and let the tool reduce it. You may also enter known values from a textbook problem. If two values are known, the missing value is found. If all three values are entered, the calculator checks consistency. This is helpful for homework, revision, and quick verification.
Matrix interpretation
A matrix with m rows and n columns represents a linear map from an n dimensional domain into an m dimensional codomain. The number of pivot columns gives the rank. Columns without pivots give free variables. Those free variables form the nullity. A full column rank matrix has nullity zero. That means no nonzero input maps to zero.
Practical learning value
The theorem also explains why some systems have many solutions. When nullity is positive, free variables exist. Each free variable adds a direction to the solution space. When rank is low, the transformation loses information. This page shows those facts with clear steps. The downloadable reports make it easy to keep records. Use the RREF display to review pivots and free variables. Then compare the result with the theorem formula.
FAQs
What does rank mean?
Rank is the number of independent output directions. For a matrix, it equals the number of pivot columns after row reduction. It also equals the dimension of the image.
What does nullity mean?
Nullity is the dimension of the kernel. It counts how many independent input directions map to zero. In matrix terms, it equals the number of free variables.
What is the rank-nullity theorem?
The theorem says rank plus nullity equals the dimension of the domain. For an m by n matrix, the domain dimension is n.
Can I calculate nullity from rank?
Yes. Subtract rank from the domain dimension. For a matrix with n columns, use nullity equals n minus rank.
Can the nullity be negative?
No. Nullity cannot be negative. If the result is negative, the entered values are inconsistent or the rank is larger than the domain dimension.
What is left nullity?
Left nullity is the dimension of the null space of the transpose. For an m by n matrix, it equals m minus rank.
Why does the calculator use row reduction?
Row reduction reveals pivot columns. Pivot columns determine rank. Once rank is known, the theorem gives nullity from the domain dimension.
What input format should I use?
Enter each matrix row on a new line. Separate values with spaces, commas, or tabs. You may also separate rows with semicolons.