null space

Thân thiện

The null space of the equation is represented by a single point at the origin.

Definition

1. Noun:
   • A space that contains no points; an empty space: In mathematics, particularly linear algebra, the null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. More generally, it can refer to a conceptual space containing nothing.

Usage

• The term is primarily used in technical, mathematical contexts to describe a specific set of solutions or a void.
• It can be used metaphorically in other fields (e.g., philosophy, physics) to discuss emptiness or the absence of elements.

Examples

• Noun:
  • To solve the system of equations, we first need to find the null space of the coefficient matrix.
  • The concept of a null space is fundamental to understanding linear transformations.
  • The data returned a null space, indicating no valid solutions existed within the given parameters.

Advanced Usage

• "To compute/find the null space": A common phrase in mathematics meaning to determine all vectors that satisfy the homogeneous equation Ax = 0.
  • The algorithm's first step is to compute the null space of the Jacobian.
• "Dimension of the null space": Refers to the number of linearly independent vectors in the null space, also known as the nullity.
  • The dimension of the null space reveals the degree of freedom in the system.

Variants and Related Words

• Nullity (n): The dimension of the null space of a linear transformation.
  • The rank-nullity theorem connects the rank of a matrix to its nullity.
• Kernel (n): A term synonymous with "null space" in linear algebra.
  • The kernel of the transformation was non-trivial.

Synonyms

• Kernel: (Mathematics) The set of elements mapped to the zero element.
• Solution space for the homogeneous equation: A descriptive synonym in linear algebra.

Related Phrases

• Left null space: The null space of the transpose of a given matrix.
  • The left null space provides information about the linear dependencies among the rows.

Học thuật

Noun

1. a space that contains no points; and empty space