Null Space
Find the null space (kernel), nullity, and determinant of any 2x2 matrix instantly. Free online linear algebra tool with basis vectors and step-by-step results for students.
About This Calculator
The null space calculator computes the kernel of any 2x2 matrix -- the set of all vectors that map to zero. Enter the four entries of your matrix and instantly get the null space description, a basis vector for the null space, the nullity (dimension), and the determinant. This tool is designed for linear algebra students, engineers, data scientists, and anyone studying vector spaces and matrix theory.
The null space (or kernel) of a matrix A is the set of all vectors v such that Av = 0. For a 2x2 matrix [[a11, a12], [a21, a22]], the null space is found by solving the system of linear equations a11x1 + a12x2 = 0 and a21x1 + a22x2 = 0. The calculator first evaluates the determinant det(A) = a11a22 - a12a21. If the determinant is non-zero, the only solution is the trivial zero vector. If the determinant is zero, the matrix rows are linearly dependent and there are infinitely many solutions forming a 1-dimensional subspace (a line through the origin). If the entire matrix is the zero matrix, every vector in R^2 is in the null space (nullity = 2).
The nullity of the matrix -- the dimension of the null space -- relates to the rank through the rank-nullity theorem: rank(A) + nullity(A) = 2 (the number of columns). A non-zero determinant implies rank 2 and nullity 0. A zero determinant with at least one non-zero entry implies rank 1 and nullity 1. The zero matrix has rank 0 and nullity 2.
Frequently Asked Questions
What is the null space of a matrix?
The null space (or kernel) of a matrix A is the set of all vectors v such that Av = 0. For a 2x2 matrix [[a,b],[c,d]], the null space consists of all vectors (x1,x2) satisfying ax1 + bx2 = 0 and cx1 + dx2 = 0. It is always a subspace of R^2 and always contains the zero vector.
How do you find the null space of a 2x2 matrix?
First compute the determinant det(A) = ad - bc. If det(A) ≠ 0, the only solution is the zero vector (nullity = 0). If det(A) = 0, find a non-zero row and solve for one variable in terms of the other. Express the solution as a scalar multiple of a basis vector -- this gives the null space basis. If the entire matrix is zero, the null space is all of R^2 (nullity = 2).
What does the nullity of a matrix mean?
The nullity of a matrix is the dimension of its null space. It represents the number of free variables when solving Av = 0. For a 2x2 matrix, nullity can be 0 (only zero vector), 1 (a line through the origin), or 2 (the entire plane). By the rank-nullity theorem, rank + nullity = number of columns.
What is the relationship between nullity and determinant?
A non-zero determinant means the matrix is invertible (full rank) and the null space contains only the zero vector (nullity = 0). A zero determinant means the matrix is singular and has a non-trivial null space (nullity >= 1). The determinant being zero is a necessary and sufficient condition for a non-trivial null space in square matrices.
Can the null space be larger than one dimension?
For a 2x2 matrix, the maximum nullity is 2, which occurs only when the matrix is the zero matrix (all entries are zero). In this case every vector in R^2 maps to zero. For any other 2x2 matrix with zero determinant, the nullity is 1, meaning the null space is a 1-dimensional line through the origin.
Is this null space calculator free to use?
Yes, this null space calculator is completely free to use with no registration or login required. You can compute as many matrices as you need.
Can I share my null space calculation results?
Yes, you can share your results by copying the page URL after calculation. The matrix values are saved in the URL, so anyone with the link will see the same computation.