Exercise 20. We define an inner product on a complex vector space V as a function that associates a complex number u,v to each pair of vectors u,v in V, such that the folowing axioms hold for all vectors u,v,w in V and all scalars k in C : 1. u+v,w=u,w+v,w 2. ku,v=ku,v 3. u,v=v,u 4. u,u0 and the equality holds if and only if u is the zero vector Prove that the following properties follow from the axioms: (a) u,uR for all uV (b) u,v+w=u,v+u,w for all u,v,wV (c) u,kv=ku,v for all u,vV and kC. Let now M22(C) be the space of 22 complex matrices, and consider the following map ,:M22(C)M22(C)(U,V)Ctr(VU) where V=VT. (d) Prove that this map is an inner product in M22(C). (e) Prove that the following standard isomorphism: T:C4(a,b,c,d)TM22(C)(acbd) is an inner product preserving isomorphism, where the inner product of C4 is given by u,v=vu Exercise 20. We define an inner product on a complex vector space V as a function that associates a complex number u,v to each pair of vectors u,v in V, such that the folowing axioms hold for all vectors u,v,w in V and all scalars k in C : 1. u+v,w=u,w+v,w 2. ku,v=ku,v 3. u,v=v,u 4. u,u0 and the equality holds if and only if u is the zero vector Prove that the following properties follow from the axioms: (a) u,uR for all uV (b) u,v+w=u,v+u,w for all u,v,wV (c) u,kv=ku,v for all u,vV and kC. Let now M22(C) be the space of 22 complex matrices, and consider the following map ,:M22(C)M22(C)(U,V)Ctr(VU) where V=VT. (d) Prove that this map is an inner product in M22(C). (e) Prove that the following standard isomorphism: T:C4(a,b,c,d)TM22(C)(acbd) is an inner product preserving isomorphism, where the inner product of C4 is given by u,v=vu