Angle between vectors and using the Cauchy-Schwarz inequalityConsider R 2 with the inner product generated by the matrix A = [ 2 1 1 1 ] .(a) Find the angle between the vectors u → = ( 3 , 3 ) and v → = ( 5 , − 8 ).(b) Show that ( v → T A T A u → ) 2 ≤ ( v → T A T A u → ) ( v → T A T A v → ) either by direct calculation or by using the Cauchy-Schwarz inequality.For part (a), how would I find the angle between the vector u → and v → ? I'm not sure how to incorporate the inner product in matrix A.For part (b), how would I use the inequality theorem to prove that statement?

Question

Angle between vectors and using the Cauchy-Schwarz inequalityConsider             R        2   with the inner product generated by the matrix   A  =      [                            2                          1                                      1                          1                      ]  .(a) Find the angle between the vectors             u      →        =  (  3  ,  3  ) and             v      →        =  (  5  ,  −  8  ).(b) Show that   (                    v        →              T        A    T    A            u      →            )    2    ≤  (                    v        →              T        A    T    A            u      →        )  (                    v        →              T        A    T    A            v      →        ) either by direct calculation or by using the Cauchy-Schwarz inequality.For part (a), how would I find the angle between the vector             u      →        and             v      →      ? I&#039;m not sure how to incorporate the inner product in matrix A.For part (b), how would I use the inequality theorem to prove that statement?

Kash Osborn · Accepted Answer

Step 1CS inequality is   &amp;lt;  u  ,  v  &amp;gt;≤  |  u  |  .  |  v  |, or                     &amp;lt;        u        ,        v        &amp;gt;                    |        u        |        .        |        v        |              ≤  1. Indeed,                     &amp;lt;        u        ,        v        &amp;gt;                    |        u        |        .        |        v        |             is the cosinus of the angle of the pair of vectors u,v.Step 2Apply this to Au,Av, you find the cos of the angle (this solve a), and as this cosinus is   ≤  1 you have proven the CS inequality, or b.

Consider R^2 with the inner product generated by the matrix A=[[2,1],[1,1]].

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