Question

Prove that every bilinear transformation maps circles vor straight line in to circle or straight line

Answer 1

Sorry I am teacher of physics

Answer 2

Every bilinear transformation maps circles or straight line in to circle or straight line

Step-by-step explanation:

Proof:  

• Let w = f(z) = (az +b)/ (cz+ d),  

• ad − bc 6 ≠ 0 be a bilinear transformation. If c = 0, then  

• f(z) = (a /d )/z + b /d  = Az + B, A = a /d and B = b /d.  

• Clearly, Az + B, being linear, maps circles and lines into circles and lines.  

• If c 6= 0, then  

• f(z) = ((a /c) *(cz + d) − (ad /c) + b) /(cz + d)  

            = (a /c )+ (bc − ad) /c ²  * 1 /z + d/c.  

• Assigning  

        z1 = z + d/c, z2 = 1 /z1 , z3 = (bc − ad /c ² )*z2  

• we obtain f(z) = a      c + z3.

• It is clear that the above transformations are of the form  
• w1 = z + α, w2 = 1 /z , w3 = βz.  
• This establishes the fact that every bilinear transformation is the resultant of bilinear.    
• ∴This proves the theorem.