Null Space Calculator
This calculator finds the null space of a small matrix. It supports two by two and two by three matrices. The null space collects every vector the matrix sends to zero. Results include a basis, the rank and the nullity. The method reduces the matrix to row echelon form. Students can check homework answers in a few seconds. Clear basis vectors make the solution easy to verify by hand.
What the Null Space Means
The null space is also called the kernel. It contains every vector x where the matrix times x equals zero. The zero vector always belongs to every null space. Interesting null spaces contain many more vectors besides zero. Each basis vector describes one free direction inside it. Any combination of basis vectors also lies inside it. Geometrically, these vectors collapse onto the origin under the matrix. That collapsing reveals what information the matrix loses.
Rank, Nullity and Free Variables
Rank counts the independent rows or pivot columns. Nullity counts the dimension of the null space. Rank plus nullity always equals the number of columns. This fact is the famous rank nullity theorem. Free variables correspond to columns without pivots. Each free variable contributes one basis vector. A full rank two by three matrix has nullity one. A rank one two by two matrix also has nullity one.
How Reduced Row Echelon Form Helps
Row reduction simplifies the system without changing its solutions. Swapping rows and scaling rows are allowed operations. Adding a multiple of one row to another is allowed too. The reduced form exposes pivot and free columns clearly. Pivot variables get expressed through the free variables. Setting one free variable to one builds a basis vector. Setting the others to zero keeps each vector independent. The calculator follows exactly this standard classroom method.
Reading a Basis Vector
A basis vector lists one value for each unknown. For two by three matrices, vectors have three entries. For two by two matrices, vectors have two entries. Multiplying the matrix by a basis vector gives zero. That multiplication is the fastest way to verify results. Basis vectors are not unique across different methods. Scaled versions of a basis describe the same space. Any valid basis spans exactly the same null space.
When Only the Zero Vector Works
Some matrices send only zero itself to zero. Their null space contains no other vectors at all. This happens when rank equals the number of columns. A two by two identity matrix is the simplest case. Its rank is two and its nullity is zero. The calculator then reports an empty basis list. That empty result is a complete and correct answer. Such matrices are called injective on their domain.
How to Use the Calculator
Six entry boxes cover both supported matrix sizes. For two by two work, the third column entries still need zeros.
- Choose 2 x 3 or 2 x 2 from the size selector.
- Enter the first row entries from left to right.
- Enter the second row entries from left to right.
- For 2 x 2 matrices, type zero in both third column boxes.
- Press Calculate to reduce the matrix automatically.
- Read rank, nullity and each basis vector shown.
Worked Example 1: A Two by Three Matrix With Rows 1 2 3 and 4 5 6
Inputs
The first example uses a two by three matrix. The first row reads one, two and three. The second row reads four, five and six. Its size selector is set to two by three. All six entries are filled before calculating.
Calculation
Row reduction produces pivots in columns one and two. The rank is therefore two for this matrix. Column three is free, so nullity equals one. The reduced rows give first variable equal to third variable. They also give second variable equal to minus twice the third.
Result
The basis contains the single vector (1, -2, 1). Rank is two and nullity is one as expected. Check the first row: one minus four plus three equals zero. Check the second row: four minus ten plus six equals zero. Both checks confirm the basis vector is correct.
Worked Example 2: A Two by Two Matrix With Rows 1 2 and 2 4
Inputs
The second example uses a two by two matrix. The first row reads one and two. Its second row reads two and four. The second row is exactly double the first row. Third column boxes hold zero, as instructed.
Calculation
Row reduction clears the second row completely. Only one pivot remains, so the rank is one. Column two is free, giving nullity of one. The reduced row states first variable equals minus twice the second. Setting the free variable to one builds the basis.
Result
The basis contains the single vector (-2, 1). Rank is one and nullity is one here. Check row one: minus two plus two equals zero. Check row two: minus four plus four equals zero. Dependence between rows created this non trivial null space.
Practical Tips for Better Results
Careful entry keeps basis vectors correct and checkable. These habits mirror good classroom practice.
- Copy matrix entries row by row, never column by column.
- Enter zeros in unused third column boxes for 2 x 2 work.
- Multiply your basis vector back through the matrix as a check.
- Remember rank plus nullity must equal the column count.
- Treat very small decimals as zero when checking by hand.
- Note that scaled basis vectors describe the same space.
Extra Notes on Null Spaces - Part 1
This check on a null space calculation keeps small errors from growing into larger problems later on. A careful review of vectors that a matrix sends to zero helps you compare options with calm and clear judgement. For many people rely on a null space calculation when they want a quick answer they can trust. When you check the inputs for a null space calculation, one wrong digit can change the final answer. Each result for vectors that a matrix sends to zero makes more sense when you know the units and limits involved. Your notes on a null space calculation stay useful when you record the date and the source figures. Using this approach works well for vectors that a matrix sends to zero because it follows the same steps every single time. In practice the result for a null space calculation should always be read as an estimate, not a promise. With care you share vectors that a matrix sends to zero with others, plain rounding makes the figures easier to compare fairly.
Extra Notes on Null Spaces - Part 2
Before you start a null space calculation, write down the question you actually need the answer to answer. One useful habit for vectors that a matrix sends to zero is checking the answer once with rough mental maths as well. Good records about a null space calculation help students, families and teams learn from earlier worked examples. This check on a null space calculation keeps small errors from growing into larger problems later on. A careful review of vectors that a matrix sends to zero helps you compare options with calm and clear judgement. For many people rely on a null space calculation when they want a quick answer they can trust. When you check the inputs for a null space calculation, one wrong digit can change the final answer. Each result for vectors that a matrix sends to zero makes more sense when you know the units and limits involved. Your notes on a null space calculation stay useful when you record the date and the source figures.
Extra Notes on Null Spaces - Part 3
Using this approach works well for vectors that a matrix sends to zero because it follows the same steps every single time. In practice the result for a null space calculation should always be read as an estimate, not a promise. With care you share vectors that a matrix sends to zero with others, plain rounding makes the figures easier to compare fairly. Before you start a null space calculation, write down the question you actually need the answer to answer. One useful habit for vectors that a matrix sends to zero is checking the answer once with rough mental maths as well. Good records about a null space calculation help students, families and teams learn from earlier worked examples. This check on a null space calculation keeps small errors from growing into larger problems later on. A careful review of vectors that a matrix sends to zero helps you compare options with calm and clear judgement. For many people rely on a null space calculation when they want a quick answer they can trust.
Extra Notes on Null Spaces - Part 4
When you check the inputs for a null space calculation, one wrong digit can change the final answer. Each result for vectors that a matrix sends to zero makes more sense when you know the units and limits involved. Your notes on a null space calculation stay useful when you record the date and the source figures. Using this approach works well for vectors that a matrix sends to zero because it follows the same steps every single time. In practice the result for a null space calculation should always be read as an estimate, not a promise. With care you share vectors that a matrix sends to zero with others, plain rounding makes the figures easier to compare fairly. Before you start a null space calculation, write down the question you actually need the answer to answer. One useful habit for vectors that a matrix sends to zero is checking the answer once with rough mental maths as well. Good records about a null space calculation help students, families and teams learn from earlier worked examples.
Extra Notes on Null Spaces - Part 5
This check on a null space calculation keeps small errors from growing into larger problems later on. A careful review of vectors that a matrix sends to zero helps you compare options with calm and clear judgement. For many people rely on a null space calculation when they want a quick answer they can trust. When you check the inputs for a null space calculation, one wrong digit can change the final answer. Each result for vectors that a matrix sends to zero makes more sense when you know the units and limits involved. Your notes on a null space calculation stay useful when you record the date and the source figures. Using this approach works well for vectors that a matrix sends to zero because it follows the same steps every single time. In practice the result for a null space calculation should always be read as an estimate, not a promise. With care you share vectors that a matrix sends to zero with others, plain rounding makes the figures easier to compare fairly.
Extra Notes on Null Spaces - Part 6
Before you start a null space calculation, write down the question you actually need the answer to answer. One useful habit for vectors that a matrix sends to zero is checking the answer once with rough mental maths as well. Good records about a null space calculation help students, families and teams learn from earlier worked examples. This check on a null space calculation keeps small errors from growing into larger problems later on. A careful review of vectors that a matrix sends to zero helps you compare options with calm and clear judgement. For many people rely on a null space calculation when they want a quick answer they can trust.
Frequently Asked Questions
Null Space Basics
1. What is a null space?
It is the set of vectors a matrix sends to zero. Mathematicians also call this set the kernel. The zero vector always belongs to it. Basis vectors describe all of its other members.
2. What does nullity mean?
Nullity is the dimension of the null space. It equals columns minus rank for any matrix. One free variable gives nullity of one. Zero nullity means only the zero vector works.
3. What is a basis vector here?
It is one independent direction inside the null space. Every solution is a combination of basis vectors. One free variable produces one basis vector. Multiplying it by the matrix gives zero.
4. Which matrix sizes are supported?
This tool supports two by two matrices. It also supports two by three matrices. Both sizes have exactly two rows. Larger systems need a more general solver.
5. Why is the zero vector always included?
Any matrix times the zero vector gives zero. That fact holds for every matrix size. Non trivial null spaces contain more vectors. Questions usually ask for those extra vectors.
Rank and Reduction Questions
6. How is rank calculated?
Rank counts pivots after row reduction finishes. It also counts independent rows in the matrix. Dependent rows add nothing to the rank. Two rows can give rank zero, one or two.
7. What is reduced row echelon form?
It is the simplest form reached by row operations. Pivots become one with zeros around them. Free columns stand out clearly in this form. Solutions read directly from the reduced rows.
8. What are free variables?
They are variables in columns without pivots. You may choose their values freely. Each choice builds solution vectors systematically. Setting one to one creates a basis vector.
9. Can rank exceed two here?
No, two rows limit rank to at most two. Rank also cannot exceed the column count. So rank stays between zero and two. Nullity adjusts to balance the column total.
10. What if both rows are zero?
Rank becomes zero for the all zero matrix. Every vector then lies in the null space. Nullity equals the full number of columns. Basis vectors become the standard unit vectors.
Checking and Using Results
11. How do I verify a basis vector?
Multiply the original matrix by the vector. Every entry of the product should be zero. Try basis vector (1, -2, 1) on the first example. Both row checks give exactly zero there.
12. Are basis vectors unique?
No, many valid bases describe one null space. Scaling a basis vector keeps it valid. Different reduction orders can produce different bases. They all span the identical solution set.
13. What does an empty basis mean?
It means only the zero vector solves the system. Nullity is zero in that situation. The identity matrix gives this classic result. Nothing is missing from such an answer.
14. Where is null space used?
Engineers use it for equilibrium and circuit analysis. Computer graphics uses it in least squares problems. Data science meets it in regression diagnostics. It reveals directions a system cannot detect.
15. Can fractions appear in basis vectors?
Yes, some matrices produce fractional entries. This tool rounds entries to four decimal places. Fractions and decimals describe identical vectors. Verify by multiplication if a value looks unusual.
CONCLUSION
This calculator reduces small matrices and reports their null spaces clearly. Rank, nullity and basis vectors appear together for easy checking. Multiply any basis vector back through the matrix to verify it. Free variables explain exactly where each basis vector comes from. Use these checks to build real confidence in linear algebra work.