Math 2.0

statement proposition

Dimension of a direct product

Statement

Let FF be a field and let VV and WW be finite-dimensional vector spaces over FF. Then the product V×WV \times W is finite-dimensional, and dim⁡(V×W)=dim⁡V+dim⁡W,\dim(V \times W) = \dim V + \dim W, where dim⁡\dim is the dimension.

Remarks

Let (vi)(v_i) be a basis of VV and let (wj)(w_j) be a basis of WW. Then the vectors (vi,0)(v_i, 0) and (0,wj)(0, w_j) together form a basis of V×WV \times W. For example, dim⁡(R2×R3)=5\dim(\R^2 \times \R^3) = 5. By induction, dim⁡(V1×⋯×Vk)=dim⁡V1+⋯+dim⁡Vk\dim(V_1 \times \cdots \times V_k) = \dim V_1 + \cdots + \dim V_k.

Explanations

L1 human

A vector of V×WV \times W is a pair, so we need the directions of VV and the directions of WW. The dimensions add up. The plane is the product of two lines, and 1+1=21 + 1 = 2.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionProduct of vector spaces, definition, no evidence levelProduct of vector spa…definitionLinear 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 direct product, statement, no evidence levelDimension of a direct…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. Product of vector spaces definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Finite-dimensional vector space definition
  10. Dimension definition
  11. Dimension of a direct product statement

Sources

  1. Web https://en.wikipedia.org/wiki/Direct_sum_of_modules, Properties

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