Math 2.0

Linear and multilinear algebra; matrix theory Definition

Row space

Builds on Linear spanMatrix

Definition

Let FF be a field and let AA be an m×nm \times n matrix over FF with rows r1,…,rm∈Fnr_1, \dots, r_m \in F^n. The row space of AA is Row⁡(A)=span⁡(r1,…,rm)⊆Fn,\operatorname{Row}(A) = \operatorname{span}(r_1, \dots, r_m) \subseteq F^n, the linear span of the rows.

Remarks

The row space of AA is the column space of ATA^T. Elementary row operations do not change the row space, so row equivalent matrices have the same row space. The nonzero rows of a row echelon form of AA form a basis of Row⁡(A)\operatorname{Row}(A). Its dimension equals the dimension of the column space, see row rank equals column rank.

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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 spandefinitionRow space, definition, no evidence levelRow 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. Row space definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Row_and_column_spaces, Row space
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