Nullspace vs. Row Space

Attribute	Nullspace	Row Space
Definition	Set of all vectors that map to the zero vector when multiplied by a given matrix	Set of all linear combinations of the rows of a matrix
Dimension	Number of columns in the matrix minus the rank of the matrix	Number of linearly independent rows in the matrix
Orthogonality	Nullspace vectors are orthogonal to row space vectors	Row space vectors are orthogonal to nullspace vectors
Intersection	Nullspace and row space can intersect at the zero vector	Row space and nullspace can intersect at the zero vector

Definition

Nullspace and row space are two fundamental concepts in linear algebra that are closely related to the properties of matrices. The nullspace of a matrix A, denoted as N(A), is the set of all vectors x such that Ax = 0. In other words, it consists of all solutions to the homogeneous equation Ax = 0. On the other hand, the row space of a matrix A, denoted as R(A), is the subspace spanned by the rows of A. It is the set of all linear combinations of the rows of A.

Dimension

One key difference between nullspace and row space is their dimension. The dimension of the nullspace of a matrix A is also known as the nullity of A, denoted as nullity(A). It is the number of linearly independent vectors in the nullspace of A. On the other hand, the dimension of the row space of a matrix A is known as the rank of A, denoted as rank(A). The rank of A is the number of linearly independent rows in A.

Relationship to the Matrix

Another important distinction between nullspace and row space is their relationship to the original matrix A. The nullspace of a matrix A is the set of all solutions to the equation Ax = 0, where x is a vector. This means that the nullspace is directly related to the properties of the matrix A itself. On the other hand, the row space of a matrix A is the subspace spanned by the rows of A, which means it is more closely tied to the rows of the matrix rather than the matrix as a whole.

Basis

Both nullspace and row space have bases that can be used to describe their respective subspaces. The basis of the nullspace of a matrix A consists of the linearly independent vectors that span the nullspace. These vectors are the solutions to the homogeneous equation Ax = 0. Similarly, the basis of the row space of a matrix A consists of the linearly independent rows of A that span the row space. These rows form a basis for the row space of A.

Orthogonality

One interesting property of nullspace and row space is their relationship to each other in terms of orthogonality. The nullspace of a matrix A is orthogonal to the row space of A. This means that any vector in the nullspace is orthogonal to any vector in the row space. This property is a consequence of the fact that the dot product of any vector in the nullspace with any vector in the row space is zero.

Applications

Nullspace and row space have important applications in various fields, including computer graphics, robotics, and signal processing. In computer graphics, the nullspace of a matrix can be used to determine the possible transformations that can be applied to an object without changing its appearance. In robotics, the row space of a matrix can be used to analyze the motion of robotic arms and determine their range of motion. In signal processing, both nullspace and row space are used to analyze and manipulate signals in various ways.