Linear Transformation and Inner Product Problem

[i_hate_math]

Homework Statement

1. Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.)
   
   PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS

Homework Equations

T(v)=A_T*v

The Attempt at a Solution

I was able to prove part a. I let v=(v1,v2) and w=(w1,w2)
apply T(v)=A_T*v and do the expansions easily yields the result

But for part b I am clueless. Please suggest an attempt

 

[geoffrey159]

Hint: the expression 'standard representation of $T$' means 'representation of $T$ in an orthonormal basis'.

 

[ehild]

You can choose any pair of independent vectors u and v to find the matrix elements of the transformation T. Let be these two vectors the base vectors of R2, (1,0) and (0,1).