Math 2.0

Linear and multilinear algebra; matrix theory Definition

Column space

Builds on Linear spanMatrix

Definition

Let FF be a field and let AA be an m×nm \times n matrix over FF with columns a1,…,an∈Fma_1, \dots, a_n \in F^m. The column space of AA is Col⁡(A)=span⁡(a1,…,an)⊆Fm,\operatorname{Col}(A) = \operatorname{span}(a_1, \dots, a_n) \subseteq F^m, the linear span of the columns.

Remarks

The column space is {Ax:x∈Fn}\{Ax : x \in F^n\}, the image of the map x↦Axx \mapsto Ax. The system Ax=bAx = b is consistent if and only if b∈Col⁡(A)b \in \operatorname{Col}(A). The dimension of the column space is the rank of AA. The pivot columns of AA form a basis of Col⁡(A)\operatorname{Col}(A).

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear span, definition, no evidence levelLinear spandefinitionColumn space, definition, no evidence levelColumn spacedefinition
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  1. Matrix definition
  2. Vector space definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear span definition
  6. Column space definition

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Row_and_column_spaces, lead section
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