Null Space Calculator

What this calculator does

The null space of a matrix is the collection of all the vectors it flattens to zero. Feed the matrix one of these vectors and the result is the zero vector. This finds that whole collection, for matrices up to 4 rows by 3 columns.

Enter the matrix and it returns a description of its null space. It runs right here in the browser.

Using the calculator

1. Set the number of rows and columns.
2. Enter the matrix's numbers in the boxes that appear.
3. Press Calculate.

It returns the vectors that describe the null space. Reset clears everything.

What the null space is

Written formally, the null space of a matrix A is every vector x for which A times x equals the zero vector. It is also called the kernel. Think of the matrix as a transformation: the null space is the set of directions that get crushed to nothing when the transformation is applied.

Every matrix has at least the zero vector in its null space, because sending zero to zero is automatic. The interesting question is whether there is anything else. Sometimes the zero vector is the only member, a "trivial" null space. Other times a whole line, or a whole plane, of vectors all collapse to zero, and that tells you the matrix is squashing space, losing information as it goes.

How it is found

Finding the null space means solving A times x equals zero, and the standard way to do that is row reduction, the same reduced row echelon form used to solve any system. Once the matrix is reduced, the columns split into two kinds: those with a pivot, and those without. The columns without a pivot mark the free variables, the ones you are free to choose, and every other variable is then forced in terms of them. Writing the solution out in terms of those free choices gives you the whole null space.

Describing it with a basis

A null space can contain infinitely many vectors, so you do not list them all. Instead you give a basis: a small set of vectors whose combinations produce every vector in the null space. If there is one free variable, the basis is a single vector, and the null space is the line through it. Two free variables give a basis of two vectors, and the null space is the plane they span. The number of vectors in the basis is called the nullity, the dimension of the null space.

The rank-nullity theorem

There is a lovely accounting rule tying this together. The rank of a matrix is the number of pivots, the count of genuinely independent rows or columns. The theorem says:

rank + nullity = number of columns

Every column is either a pivot column, adding to the rank, or a free column, adding to the nullity, so the two must sum to the total number of columns. It means a bigger null space always comes at the cost of a lower rank. The more directions a matrix crushes, the fewer it keeps.

A worked example

Take the matrix [[1, 2], [2, 4]]. Its second row is just twice its first, so the rows are not independent. Solving A times (x, y) equals zero gives the single equation x + 2y = 0, since both rows say the same thing. That rearranges to x = -2y.

So any solution looks like (x, y) = y times (-2, 1), for any value of y. The null space is the line spanned by the vector (-2, 1). Checking the theorem: the rank is 1 (one pivot), the nullity is 1 (one basis vector), and 1 + 1 = 2, the number of columns.

Questions people ask

What is the null space of a matrix?

The set of all vectors x that the matrix sends to the zero vector, meaning A times x equals zero. It is also called the kernel.

Is the zero vector always in the null space?

Yes, always, because any matrix times the zero vector gives zero. The question is whether the null space contains anything more.

How do you find the null space?

Row-reduce the matrix, identify the free variables from the non-pivot columns, and express the solutions in terms of them. That gives a basis for the null space.

What is the rank-nullity theorem?

It states that the rank plus the nullity equals the number of columns. Each column either contributes a pivot to the rank or a free variable to the nullity.

What does a nonzero null space mean?

That the matrix crushes some nonzero directions to zero, losing information. Such a matrix is singular and cannot be inverted.

References

A note on where this comes from. The null space, or kernel, of a matrix is the solution set of the homogeneous equation A times x equals zero, found by reducing the matrix to row echelon form. Its dimension, the nullity, satisfies the rank-nullity theorem, a fundamental result of linear algebra. For further reading, see Kernel (linear algebra).

1. The null space (kernel), the set of vectors x with A times x equal to zero, described by a basis.
2. The rank-nullity theorem, rank plus nullity equals the number of columns.