Math 2.0

definition

Linear subspace

Definition

Let FF be a field and let VV be a vector space over FF. A linear subspace of VV is a subset W⊆VW \subseteq V with these three properties:

  1. 0∈W0 \in W.
  2. If u,v∈Wu, v \in W, then u+v∈Wu + v \in W.
  3. If a∈Fa \in F and v∈Wv \in W, then av∈Wav \in W.

Remarks

With the operations of VV, a subspace WW is itself a vector space over FF. The sets {0}\{0\} and VV are subspaces of VV. The lines through the origin are the one-dimensional subspaces of R2\R^2; a line that misses the origin is not a subspace. The kernel and the image of a linear map are subspaces.

Explanations

L1 human

A subspace is a smaller vector space inside a bigger one. We cannot leave it by adding its vectors or by scaling them. In space, the subspaces are the origin, the lines and planes through the origin, and the whole space.

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Vector space, definition, no evidence levelVector spacedefinitionLinear subspace, definition, no evidence levelLinear subspacedefinition
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  1. Vector space definition
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Sources

  1. Web https://en.wikipedia.org/wiki/Linear_subspace, Definition

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