Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

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SUMMARY

The discussion centers on proving the dimension formula for subspaces U and W of a vector space V, specifically that dim(U + W) = dim(U) + dim(W) - dim(U ∩ W). Participants emphasize the importance of visualizing the relationship between the subspaces using a Venn diagram, which illustrates that elements in the intersection U ∩ W are counted twice when summing the dimensions of U and W. To formalize the proof, it is essential to construct a basis for U ∩ W and then extend it to bases for U and W, allowing for a clear demonstration of the dimension relationship.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with the concept of dimension in linear algebra
  • Knowledge of basis vectors and their properties
  • Ability to interpret Venn diagrams in the context of set theory
NEXT STEPS
  • Study the properties of vector space bases and dimension
  • Learn about the intersection of subspaces and its implications
  • Explore the concept of direct sums in linear algebra
  • Practice proving dimension formulas for various combinations of subspaces
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Students and educators in linear algebra, mathematicians focusing on vector spaces, and anyone seeking to deepen their understanding of dimension theory in mathematics.

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Homework Statement


Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W).


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The Attempt at a Solution


I can see this picture-wise in a venn-diagram form. In adding U and W you count the elements in their intersection twice, since both spaces contain them. Thus, you subtract the dimension of U intersect W. But, how do you show this in proof format? It doesn't seem like a verbal explanation is enough.
 
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What you really are thinking about in your Venn diagram are basis vectors. Write a basis for U intersect W. Extend it to a basis for U and W. Now do the Venn argument.
 

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