I have a linear map R α = ( c o s α − s i n α s i n α c o s α ) and the question goes like, how does R α transform a vector?Can someone please help me?

Question

I have a linear map       R    α    =      (                            c          o          s          α                          −          s          i          n          α                                      s          i          n          α                          c          o          s          α                      )   and the question goes like, how does       R    α   transform a vector?Can someone please help me?

nezivande0u · Accepted Answer

The comments above on the OP are good, but here&#039;s another perspective. How does       R    α   change the length of a vector? It is easy to verify that      R    α    T        R    α    =      (                                        cos            2                    ⁡          α          +                      sin            2                    ⁡          α                          0                                      0                                      cos            2                    ⁡          α          +                      sin            2                    ⁡          α                      )    =  ISo that for some vector v we have       |        |        R    α    v      |              |        2    =  (      R    α    v      )    T    (      R    α    v  )  =      v    T        R    α    T        R    α    v  =      v    T    I  v  =      v    T    v  =      |        |    v      |              |        2  . That is,       R    α   is an isometry (it can&#039;t change the length of vectors). So if       R    α   cannot change the length of a vector, it could only possibly rotate a vector. So what is this angle? We can compute the angle,   θ, between       R    α    v and v by computing  cos  ⁡  θ  =            (              R        α            v      )      ⋅      v                      |                    |                    R        α            v              |                    |            ⋅              |                    |            v              |                    |              .I&#039;ll leave it to you to verify that this gives   θ  =  α (mod   π or whatever)

I have a linear map R_alpha=((cos alpha,-sin alpha),(sin alpha,cos alpha)) and the question goes like, how does R_alpha transform a vector?

Answered question

Answer & Explanation

New Questions in Precalculus