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Let V be the vector space of ordered pairs of complex numbers over the real field `RR`...

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gretjxfj | Student, Undergraduate | (Level 1) Honors

Posted September 4, 2013 at 7:55 AM via web

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Let V be the vector space of ordered pairs of complex numbers over the real field `RR` . Show that V is of dimension 4.

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aruv | High School Teacher | (Level 2) Valedictorian

Posted September 4, 2013 at 8:34 AM (Answer #1)

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Let S={(1,0),(0,1),(i,0),(0,i)} be the set . We wish to prove S is basis of V over the real field R.Consequently we can prove that  dim(V)=4, because number of elements in set S is 4.

 Vectors in S are linearly independent. If

a(1,0)+b(0,1)+c(i,0)+d(0,i)=0 

a+ic=0 => a=c=0

b+id=0  => b=d=0 

a=b=c=d=0   only solution.

Thus vectors in S are linearly independent.

Let (x+iy,w+iz) in V

then

(x+iy,w+iz)=x(1,0)+y(i,0)+w(0,1)+z(0,i)

(x+iy,w+iz) is an arbitrary element of V. Thus every element of V can be expressed as linear combination of vectors in S.

Thus S is basis for V. Number of elements in S  are 4 ,so dimension of V be 4.

Hence proved.

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