The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector, representing the solutions to the homogeneous equation Ax = 0. It is a fundamental concept in linear algebra, reflecting the linear dependencies among the columns of the matrix and playing a crucial role in understanding the solution space of linear systems.

Null Space

null space

set of all vectors

The zero vector is a vector in a vector space that has all of its components equal to zero, serving as the additive identity. It plays a crucial role in linear algebra as it is the only vector that, when added to any other vector, leaves the other vector unchanged.

zero vector

A homogeneous equation is a type of equation in which all terms are of the same degree, making it invariant under scaling transformations of its variables. In differential equations, a homogeneous equation often refers to one where all terms are dependent on the function and its derivatives, allowing solutions to be found using specific techniques like substitution or the method of undetermined coefficients.

homogeneous equation

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.

linear algebra

linear dependencies

columns of the matrix

Solution space refers to the set of all possible solutions to a given problem, often visualized in optimization and decision-making contexts. It is crucial for understanding the range of potential outcomes and for identifying the optimal solution based on specific criteria or constraints.

solution space

Linear systems are mathematical models of systems that can be described using linear equations, where the principle of superposition applies, meaning the output is directly proportional to the input. They are foundational in understanding and analyzing various phenomena in engineering, physics, and applied mathematics, allowing for simplified solutions and predictions of system behavior.

Your Courses

Relevant Degrees

Your Lessons