CC^n is a CC-vector space with a natural structure of RR-vector space. Then, what is the dimension of CC^n as RR-vector space? I'm struggling so hard with this question, I do not even know how to start. Can somebody help me please.

Jefferson Booth

Jefferson Booth

Answered question

2022-11-08

C n is a C-vector space with a natural structure of R -vector space. Then, what is the dimension of C n as R -vector space? I'm struggling so hard with this question, I do not even know how to start. Can somebody help me please.

Answer & Explanation

kuthiwenihca

kuthiwenihca

Beginner2022-11-09Added 23 answers

Every element v C n is uniquely expressed using 2n real numbers a 1 , , a n , b 1 , , b n R as
v = ( a 1 + i b 1 a 2 + i b 2 a n + i b n ) = a 1 ( 1 0 0 ) + + a n ( 0 0 1 ) + b 1 ( i 0 0 ) + + b n ( 0 0 i ) .
Hence, the 2n vectors
( 1 0 0 ) , , ( 0 0 1 ) , ( i 0 0 ) , , ( 0 0 i )
form a basis of C n as an R -vector space.
Yaretzi Mcconnell

Yaretzi Mcconnell

Beginner2022-11-10Added 3 answers

If { e 1 , , e n } is the standard basis of C n , consider
{ e 1 , , e n , i e 1 , , i e n }

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