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Proofs: Complex numbers and vectors

 

Complex numbers

 

 

Proposition:
For complex numbers \(z_1, z_2\):

  1. \( (z_1 + z_2)^* = z_1^* + z_2^* \)
  2. \( (z_1 z_2)^* = z_1^* z_2^* \)
  3. \( |z_1 z_2| = |z_1| |z_2| \)

Proof:
Let \(z_1 = a_1 + ib_1\) and \(z_2 = a_2 + ib_2\).

  1. $$ (z_1 + z_2)^* \eql (a_1 + ib_1 + a_2 + ib_2)^* \eql a_1+a_2 - i(b_1 + b_2) \eql (a_1 - ib_1) + (a_2 - ib_2) \eql z_1^* + z_2^* $$
  2. $$ (z_1 z_2)^* \eql ( (a_1 + ib_1)(a_2 + ib_2) )^* \eql ( a_1 a_2 - b_1 b_2 + (a_1 b_1 + a_2 b_1) i )^* \eql a_1 a_2 - b_1 b_2 - (a_1 b_1 + a_2 b_1) i ) \eql a_1 a_2 - b_1 b_2 - (a_1 b_1 + a_2 b_1) i ) \eql (a_1 - ib_1) (a_2 - ib_2) \eql z_1^* z_2^* $$
  3. $$ |z_1 z_2|^2 \eql (z_1 z_2) (z_1 z_2)^* \eql (z_1 z_2) (z_1^* z_2^*) \eql (z_1 z_z^*) (z_2 z_2^*) \eql |z_1|^2 |z_2|^2 $$ Since magnitudes are positive real numbers, \(|z_1 z_2| = |z_1| |z_2|\).
 

Complex vectors

 



© 2022, Rahul Simha