Math 2.0

statement proposition

Dimension of a subspace

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. Then WW is finite-dimensional, and dim⁡W≤dim⁡V\dim W \le \dim V, where dim⁡\dim is the dimension.

Remarks

A basis of WW is a linearly independent family in VV, so it has at most dim⁡V\dim V vectors. Equality holds only when W=VW = V, see a subspace of full dimension is the whole space. For infinite-dimensional VV, the inequality dim⁡W≤dim⁡V\dim W \le \dim V of cardinals still holds.

Explanations

L1 human

A subspace cannot have more independent directions than the space around it. A plane in ordinary space has dimension 22, which is at most 33. The only subspace with the same dimension is the whole space.

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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 levelDimensiondefinitionDimension of a subspace, statement, no evidence levelDimension of a subspa…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. Dimension of a subspace statement

Sources

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

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