Simple Inner Product Proof (complex)

[binbagsss]

I'm okay on proving the other properties, just struggling with what to do on this one:

(v,v)≥0, with equality iff v=0,where the inner product is defined as:

z1w*1+iz2w*2

(where * represent the complex conjugate)

My working so far is:
u1u*1+iu2u*2
=u1^2 + iu2^2


(I'm not sure what to do next and how to deal with the i algebriacally. I've done real ones and complex one without an i in the definition,and seem ok with this property for them).Thanks a lot, greatly appreaciated =].

 

[tiny-tim]

hi binbagsss!

(try using the X₂ button just above the Reply box )

I'm okay on proving the other properties, just struggling with what to do on this one:

(v,v)≥0, with equality iff v=0,where the inner product is defined as:

z1w*1+iz2w*2

(where * represent the complex conjugate)

does (u,v) = (v,u)* ?

 

[pasmith]

Your calculations are largely correct (you need modulus signs on u1 and u2 in your final expression).

Since $\langle (u_1,u_2),(u_1,u_2)\rangle = |u_1|^2 +i|u_2|^2$ is not necessarily real, the conclusion must be that $\langle\cdot,\cdot\rangle$ as defined is not an inner product.