Null space

Definition

The null space of a matrix is the set of all vectors that, when multiplied by that matrix, yield the zero vector. This concept is crucial for understanding solutions to linear equations, as it indicates which vectors do not change the output when applied to the transformation represented by the matrix. The null space helps in exploring the dimensions of vector spaces and the relationships between different solutions to linear systems.

5 Must Know Facts For Your Next Test

1. The null space is also referred to as the kernel and plays a vital role in solving homogeneous linear equations.
2. A matrix's null space can be computed using row reduction techniques to find its reduced row echelon form.
3. The dimension of the null space is known as the nullity of the matrix and is related to the rank-nullity theorem.
4. If a matrix has full column rank, its null space consists only of the zero vector, indicating that the system has a unique solution.
5. Understanding the null space helps determine whether a set of vectors is linearly independent or dependent.

Review Questions

• How does the null space relate to solving systems of linear equations?
  • The null space is directly related to solving homogeneous systems of linear equations, where the equation can be expressed in the form Ax = 0. Vectors in the null space represent solutions that lead to no change in output when transformed by the matrix A. Understanding these solutions helps identify if there are multiple solutions or if the solution is unique, based on whether non-zero vectors exist in this space.
• Discuss how the rank-nullity theorem connects rank and nullity for a given matrix.
  • The rank-nullity theorem states that for any matrix A, the sum of its rank (the dimension of its column space) and its nullity (the dimension of its null space) equals the number of columns in A. This relationship shows how many independent columns contribute to solutions while also revealing how many vectors lie in the null space. It emphasizes that understanding both concepts is crucial for grasping the full structure of linear transformations represented by matrices.
• Evaluate the implications of a non-trivial null space on linear independence among columns of a matrix.
  • A non-trivial null space indicates that there are non-zero vectors that, when transformed by the matrix, result in the zero vector. This implies that at least one column of the matrix can be expressed as a linear combination of others, leading to a conclusion that not all columns are linearly independent. Thus, knowing about a matrix's null space allows one to assess whether its columns form a basis for its range or if redundancy exists within those vectors.

Related terms

Kernel:

The kernel of a linear transformation is another term for the null space, representing all input vectors that map to the zero vector.

Linear Transformation:

A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Rank: The rank of a matrix is the dimension of its column space, which provides insight into the number of linearly independent columns in the matrix.