Transformation on Inner Product Space
Let be a real inner product space and there is a map which preserves the inner product, that is, for all vectors and in ,
Must be linear map?
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1 solution
A straightforward but tedious computation shows that ⟨ T ( a v + w ) − a T ( v ) − T ( w ) , T ( a v + w ) − a T ( v ) − T ( w ) ⟩ = 0 for all vectors v , w in V and all scalars a , by linearity in both arguments. Since the inner product is positive definite, it follows that T ( a v + w ) = a T ( v ) + T ( w ) , meaning that T is a linear map.