Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal projection, prove that ||T(x)||≤||x|| for all x ∈V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x∈V? (b) Suppose that T is a projection such that ||T(x)||≤||x||for x ∈V.Prove that T is an orthogonal projection.

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