Null Space of a Matrix

The null space of a matrix is the collection of all vectors that satisfy the equation Ax = 0, where A is a matrix, x is a vector, and 0 is the zero vector. It contains all vectors that become zero after being multiplied by the matrix.

For a matrix A of order m × n:

• A is an m × n matrix.
• x is an n × 1 column vector.
• 0 is an m × 1 zero vector.

The null space is an important concept in linear algebra because it helps determine the solutions of a system of linear equations. The dimension of the null space is called the nullity of the matrix, which represents the number of independent solutions to the equation Ax = 0.

Geometric Interpretation of Null Space

• The null space is a subspace of the input space (domain) of a matrix.
• For an m × n matrix A, the null space is a subspace of Rⁿ.
• It consists of all vectors that are transformed into the zero vector when multiplied by A.
• The dimension of the null space is called the nullity of the matrix.
• Nullity represents the number of independent directions in the input space that are mapped to zero by the transformation.
• Geometrically, the null space describes the vectors that lose their effect under the transformation represented by the matrix.

Steps to Find the Null Space

We need to follow a series of steps to find the Null Space of a matrix. These steps involve Row reduction and solve a system of linear equations. The steps are:

• Step 1: Write the Matrix Equation

We are given a matrix A of M × N order, write the matrix equation Ax = 0, where x is the vector [x₁ , x₂ , . . . , x_n]^T.

• Step 2: Make the Augmented Matrix

Form the Augmented Matrix [A|0], where 0 represents the zero vector of appropriate dimensions. This matrix essentially represents the system of equations Ax = 0.

• Step 3: Perform Row Reduction

Conduct a Gaussian elimination (or row reduction) on the augmented matrix to put it into Reduced Row Echelon Form (RREF). The matrix will be simplified in order for system of equations to be easily answered.

• Step 4: Identify Free and Pivot Variables

Determine the pivot columns (those containing leading ones) and the free variables (those associated with non-pivot columns) in the RREF matrix. The free variables shall serve as parameters of solutions.

• Step 5: Express the Solution in Parametric Form

Substitute pivot variables for free ones. These solutions will represent vectors situated at null space. It gives us vectorized general solution and thus allows to express a null space basis.

Properties of a Null Space

The null space has several important properties that help in understanding the structure of a matrix and the solutions of homogeneous systems of linear equations.

• The null space of a matrix is a subspace of its input vector space.
• It contains all vectors x that satisfy the equation Ax = 0.
• The dimension of the null space is called the nullity of the matrix.
• Nullity represents the number of free variables in the system Ax = 0.
• The null space is spanned by a set of linearly independent vectors called its basis.
• For an m × n matrix, the Rank-Nullity Theorem states that:Rank(A) + Nullity(A) = n

Also Read: Null Space and Nullity of a Matrix