Dimension of U,V and W over K: Do they Equal?

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SUMMARY

The discussion clarifies that having vector spaces U, V, and W defined over the same field K does not imply that their dimensions are equal. A concrete example provided is the comparison between R^2 and R^3, both of which are vector spaces over the field R, yet they have different dimensions: dim R^2 equals 2 while dim R^3 equals 3. This highlights that dimension is a property independent of the field over which the spaces are defined.

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  • Familiarity with the concept of dimension in linear algebra
  • Basic comprehension of examples involving R^n spaces
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to vector spaces and their dimensions.

garyljc
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Just have a question
if U,V and W are over the same field K
does it mean that dim U = dim V = dim W ?
 
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No.

For example, [itex]R^2[/itex] and [itex]R^3[/itex] are both over the same field R, but R^2 has dimension 2 and R^3 has dimension 3.
 

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