- #1
tandoorichicken
- 245
- 0
Is it possible for a nullspace to be spanned by only one vector? Does a statement like Nul A = Span{[itex]\vec{v}_1[/itex]} even have any meaning?
Is a Nullspace Spanned by One Vector Possible?
- Thread starter tandoorichicken
- Start date
-
- Tags
- Vector Vector spaces
Click For Summary
SUMMARY
A nullspace can indeed be spanned by a single vector, specifically when the matrix has a rank less than its number of columns. In the example provided, the nullspace of the matrix \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) is spanned by the vector \(\vec{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\). This indicates that the nullspace is defined as the set of all vectors that satisfy the equation \(A\vec{x} = \vec{0}\), where \(\vec{x}\) can be expressed as a linear combination of the spanning vector.
PREREQUISITES- Understanding of linear algebra concepts, particularly nullspaces and spans.
- Familiarity with matrix representation and operations.
- Knowledge of vector spaces and their properties.
- Ability to solve linear equations in the context of matrices.
- Study the concept of nullspace in more detail, focusing on its properties and applications.
- Learn about the rank-nullity theorem and its implications for linear transformations.
- Explore examples of nullspaces for different matrix types, including full rank and deficient rank matrices.
- Investigate the relationship between nullspaces and linear independence of vectors.
Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to clarify concepts related to vector spaces and nullspaces.
Similar threads
- · Replies 8 ·
- · Replies 17 ·
- · Replies 4 ·
- · Replies 2 ·
- · Replies 24 ·