Show that ⟨ x + y , x − y ⟩ = ‖ x ‖ 2 2 − ‖ y ‖ 2 2 ?given x , y ∈ R n I am trying to prove this: ⟨ x − y , x − y ⟩ = ⟨ x , x ⟩ − 2 ⟨ x , y ⟩ + ⟨ y , y ⟩ = ‖ x ‖ 2 2 − 2 ⟨ x , y ⟩ + ‖ y ‖ 2 2 .What I tried is using the fact that: ⟨ x − y , x − y ⟩ = ⟨ x , x ⟩ − 2 ⟨ x , y ⟩ + ⟨ y , y ⟩ = ‖ x ‖ 2 2 − 2 ⟨ x , y ⟩ + ‖ y ‖ 2 2 .Then, since y=−(−y), I have ⟨ x + y , x − y ⟩ = ⟨ x − ( − ) y , x − y ⟩But I don't know how I proceed from this... Maybe my approach is wrong?

Question

Show that   ⟨  x  +  y  ,  x  −  y  ⟩  =  ‖  x      ‖    2    2    −  ‖  y      ‖    2    2  ?given   x  ,  y  ∈            R        n   I am trying to prove this:  ⟨  x  −  y  ,  x  −  y  ⟩  =  ⟨  x  ,  x  ⟩  −  2  ⟨  x  ,  y  ⟩  +  ⟨  y  ,  y  ⟩  =  ‖  x      ‖    2    2    −  2  ⟨  x  ,  y  ⟩  +  ‖  y      ‖    2    2    .What I tried is using the fact that:  ⟨  x  −  y  ,  x  −  y  ⟩  =  ⟨  x  ,  x  ⟩  −  2  ⟨  x  ,  y  ⟩  +  ⟨  y  ,  y  ⟩  =  ‖  x      ‖    2    2    −  2  ⟨  x  ,  y  ⟩  +  ‖  y      ‖    2    2    .Then, since y=−(−y), I have  ⟨  x  +  y  ,  x  −  y  ⟩  =  ⟨  x  −  (  −  )  y  ,  x  −  y  ⟩But I don&#039;t know how I proceed from this... Maybe my approach is wrong?

escobamesmo · Accepted Answer

The simplest way to proceed is by using the bilinearity of the inner product, i.e., just expanding ⟨x+y,x−y⟩. Doing this:                    ⟨        x        +        y        ,        x        −        y        ⟩                            =        ⟨        x        ,        x        −        y        ⟩        +        ⟨        y        ,        x        −        y        ⟩                                          =        ⟨        x        ,        x        ⟩        −        ⟨        x        ,        y        ⟩        +        ⟨        y        ,        x        ⟩        −        ⟨        y        ,        y        ⟩                                          =        ‖        x                  ‖          2                −        ‖        y                  ‖          2                    since the two middle terms cancel out (they are equal, inner product is commutative).

Given x,y in RR^n I am trying to prove this: ⟨x+y,x−y⟩=∥x∥_2^2−∥y∥_2^2

Answered question

Answer & Explanation

New Questions in Precalculus