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Matrix Nullity Formula:
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Matrix nullity represents the dimension of the null space of a matrix, which is the number of linearly independent solutions to the homogeneous equation Ax = 0. It provides important information about the structure and properties of linear transformations.
The calculator uses the fundamental nullity formula:
Where:
Explanation: This formula is derived from the Rank-Nullity Theorem, which states that for any matrix A, the sum of its rank and nullity equals the number of columns.
Details: Calculating nullity is essential for understanding the solution space of linear systems, determining whether a matrix is invertible, and analyzing the properties of linear transformations in various mathematical and engineering applications.
Tips: Enter the number of columns in the matrix and the rank of the matrix. Both values must be non-negative integers, and the rank cannot exceed the number of columns.
Q1: What does nullity tell us about a matrix?
A: Nullity indicates the number of free variables in the solution to Ax = 0, which corresponds to the dimension of the kernel of the linear transformation represented by A.
Q2: When is the nullity of a matrix zero?
A: Nullity is zero when the matrix has full column rank, meaning all columns are linearly independent and the only solution to Ax = 0 is the trivial solution.
Q3: How is nullity related to invertibility?
A: A square matrix is invertible if and only if its nullity is zero, which means the matrix has full rank.
Q4: Can nullity be larger than the number of columns?
A: No, nullity cannot exceed the number of columns since it's calculated as n - rank(A), and rank(A) is always non-negative.
Q5: What's the difference between nullity and rank?
A: Rank measures the dimension of the column space (number of linearly independent columns), while nullity measures the dimension of the null space (number of solutions to Ax = 0).