Understanding Matrices and Linear Maps
A matrix is a rectangular array of numbers organised into rows and columns. When viewed as a linear transformation, a matrix describes how to move or rotate vectors in space. For instance, a 2×2 matrix can rotate or scale vectors in a plane, a 3×3 matrix operates on three-dimensional space, and a 4×4 matrix works in four-dimensional space.
The relationship between a matrix A and a vector x is central to linear algebra. When we multiply A by x, we obtain Ax—a new vector in the output space. Different input vectors produce different outputs; some map to the zero vector, and those special inputs form the null space.
The Null Space and Kernel
The null space of a matrix A is the set of all vectors x such that Ax = 0. In matrix theory, this is synonymous with the kernel of the matrix when treated as a linear map.
Unlike finite sets, the null space typically contains infinitely many vectors (unless it only contains the zero vector itself). To describe it completely, we determine a basis—a minimal set of linearly independent vectors from which every vector in the null space can be expressed as a linear combination. The basis vectors are the generators of the null space.
- Trivial null space: Contains only the zero vector (full rank matrices).
- Non-trivial null space: Contains infinitely many vectors spanning a proper subspace.
Finding the Null Space
To find the null space, we use Gauss-Jordan elimination to reduce the matrix to reduced row echelon form (RREF). This process:
- Applies elementary row operations (row swaps, scalar multiplication, and row addition).
- Converts the matrix into RREF, where pivot positions clearly identify dependent and free variables.
- Expresses free variables as parameters to construct basis vectors.
For an m × n matrix with rank r, the nullity (dimension of null space) is n − r.
Nullity = Number of Columns − Rank of Matrix
Nullity— The dimension of the null space—the number of free variables after row reduction.Rank— The number of pivot (non-zero) rows in the reduced row echelon form.Number of Columns— The number of variables (columns) in the original matrix.
Practical Example
Consider a 3×4 matrix:
⎡ 1 2 3 4 ⎤
⎢ 2 4 6 8 ⎥
⎣ 3 6 9 12 ⎦
Notice that rows 2 and 3 are multiples of row 1. After row reduction, we find rank = 1. With 4 columns, the nullity = 4 − 1 = 3. This means the null space has 3 basis vectors, and the system Ax = 0 has infinitely many solutions described by three free parameters.
Common Pitfalls and Caveats
Avoid these mistakes when working with null spaces:
- Confusing rank with dimension — The rank tells you the number of independent rows, not the dimension of the null space. Use the rank-nullity theorem: <span style="font-family:monospace">rank + nullity = number of columns</span>. A full-rank square matrix has a trivial null space containing only zero.
- Arithmetic errors in row reduction — Gauss-Jordan elimination requires precise arithmetic. A single computational mistake propagates through the remaining steps, producing incorrect basis vectors. Always double-check pivot operations and ensure you're eliminating both above and below each pivot, not just below.
- Forgetting to identify free variables — After row reduction, free variables (non-pivot columns) must be parameterised to construct basis vectors. Skipping this step leaves you with partial information. Each free variable corresponds to one dimension of the null space.
- Assuming the null space is always non-trivial — Square matrices with full rank (equal to their size) have null spaces containing only the zero vector. Not all matrices have 'interesting' null spaces; check the rank first to determine whether basis vectors exist beyond zero.