Math 2.0

Linear and multilinear algebra; matrix theory Definition

Null space

Builds on Kernel of a linear mapMatrix

Definition

Let FF be a field and let AA be an m×nm \times n matrix over FF. The null space of AA is Null⁡(A)={x∈Fn:Ax=0}.\operatorname{Null}(A) = \{x \in F^n : Ax = 0\}. It is the kernel of the linear map x↦Axx \mapsto Ax.

Remarks

The null space is a subspace of FnF^n, of dimension n−rank⁡An - \rank A, the nullity of AA. Row equivalent matrices have the same null space. The null space of (11)\begin{pmatrix} 1 & 1 \end{pmatrix} is the line spanned by (1,−1)T(1, -1)^T. Each vector of the null space is orthogonal to each row of AA, for the dot product.

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To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionNull space, definition, no evidence levelNull spacedefinition
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  1. Matrix definition
  2. Vector space definition
  3. Linear map definition
  4. Kernel of a linear map definition
  5. Null space definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Kernel_(linear_algebra), lead section
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