Mathematics for Data Science - Lecture Notes

Professor Sarang Sane, Department of Mathematics, IIT Madras

Topic: Null Space of a Matrix - Part 1

Objectives:

• Understand the concept of the null space of a matrix.
• Determine the nullity and a basis for the null space.
• Introduction to the rank-nullity theorem.
• Using determinants to verify if a set of vectors forms a basis.

Key Concepts:

Null Space of a Matrix

• Null space: The solution space for a homogeneous system of linear equations Ax = 0, where A is an m x n matrix.
• It is a subset of ( R^n ) and forms a vector space on its own.
• Why "null"?: Because it comprises vectors x such that Ax = 0.
• Dimension: The number of vectors in the basis for the null space, termed the nullity of A.

Subspace Verification

• A subspace is a vector space that satisfies:
  • If x, y are in the subspace W, then x + y is also in W.
  • If x is in W and ( \lambda ) is a scalar, then ( \lambda x ) is in W.
• Null space W of A is a vector space in its own right.

Finding Null Space and Nullity

• Use Gaussian elimination or row reduction to find solutions of Ax = 0.

Dependent and Independent Variables

• Dependent Variable: Corresponds to columns with leading entries (1's) in reduced row echelon form (RREF).
• Independent Variable: Columns without leading entries.
• Nullity: Number of independent variables in the system.

Solving Ax = 0

• Assign arbitrary values ( t_i ) to independent variables.
• Compute dependent variables using RREF equations.

Finding a Basis

• For each independent variable, set one ( t_i = 1 ) and the rest to 0.
• Resultant vectors form a basis for the null space.

Example

• Given a matrix A: [ \begin{pmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ 3 & 3 & 3 \end{pmatrix} ]
• Augmented matrix becomes: [ \begin{pmatrix} 1 & 1 & 1 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} ]
  • Dependent: x1, Independent: x2, x3.
  • Nullity: 2.
  • Basis: [ (-1, 1, 0), (-1, 0, 1) ].
  • Solution space represented by linear combinations of basis vectors.

Efficiency Consideration

• For computational efficiency, avoid writing out the augmented 0 vector for homogeneous systems during row reduction.

Conclusion

• Null space and rank-nullity concepts are essential in understanding solutions of systems of equations and their properties.
• Gaussian elimination helps find the basis and nullity of matrix null spaces efficiently.