Question

5. (a) Explain why the standard inner product is invariant under an orthogonal trans formation. That is, if U is any orthogonal miatrix, and if u = Ux and v = Uy, then i.e. multiplication by an orthogonal matrix does not change the standard inner product. (b) Given any two vectors x. y in R", explain why the angle between them is Py invarient under an orthogonal transformation. That is, if u where P is an orthogonal matrix, thern Px and v cos θυ.v-cos6,

Answer 1

For any orthogonal transformation T: Rn -->Rn, there is a n×n orthogonal matrix U such that T(x) = Ux.hence T also preseves angle between vectors.

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