Math 2.0

statement proposition

A subspace of full dimension is the whole space

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a linear subspace of VV. If dim⁡W=dim⁡V\dim W = \dim V, then W=VW = V.

Remarks

A basis of WW is a linearly independent family of dim⁡V\dim V vectors in VV. So it is a basis of VV, see linearly independent set of the right size is a basis. The statement fails for infinite-dimensional spaces. In F[x]F[x], the subspace of polynomials with p(0)=0p(0) = 0 has the same dimension as F[x]F[x], but it is not equal to F[x]F[x].

Explanations

L1 human

If a subspace is as large as the whole space, measured by dimension, then it is the whole space. For example, a two-dimensional subspace of the plane is the plane. This lets us prove that two subspaces are equal by an inclusion and a count.

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To read this node, read these 10 nodes in this order. The foundations come first, and the list ends with this node.

Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionA subspace of full dimension is the whole space, statement, no evidence levelA subspace of full di…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Spanning set definition
  7. Basis definition
  8. Finite-dimensional vector space definition
  9. Dimension definition
  10. A subspace of full dimension is the whole space statement

Sources

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties

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