Dimension of Complex Vector Spaces over R?

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SUMMARY

The discussion clarifies that a complex vector space C, which has a dimension of 1 over the complex field, has a dimension of 2 when considered as a vector space over the real numbers (R). This is because each complex dimension corresponds to two real dimensions, one for the real part and one for the imaginary part. Therefore, the correct conclusion is that the dimension of C over R is 2, not 0 or 1.

PREREQUISITES
  • Understanding of complex vector spaces
  • Knowledge of real vector spaces
  • Familiarity with the concept of dimension in linear algebra
  • Basic grasp of the relationship between complex and real numbers
NEXT STEPS
  • Study the properties of complex vector spaces in linear algebra
  • Learn about the relationship between dimensions of vector spaces over different fields
  • Explore examples of complex vector spaces and their real counterparts
  • Investigate the implications of dimensionality in higher-dimensional spaces
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Students and educators in mathematics, particularly those studying linear algebra, complex analysis, or anyone interested in the properties of vector spaces over different fields.

gtfitzpatrick
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Homework Statement


C is a 1-dim comples space. What is its dimesion when regarded a vector space over R

The Attempt at a Solution



if the complex vector space is dim 1 then the real part must = 0 so therefore when viewed over R the dimension must be 0 as well
 
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gtfitzpatrick said:
if the complex vector space is dim 1 then the real part must = 0 so therefore when viewed over R the dimension must be 0 as well

Could you elaborate on this, since I don't understand your reasoning at all...
You know that the only 0-dimensional spaces is {0}, right? So there's no way C could have dimension 0...
 
sorry, i should have said a complex vector is dim 2 when viewed over R, but in this case its dim is 1 so the real part = 0. therefore when view over R only the complex part existins so when viewed over R is dimension is also 1?
 

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