# 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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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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Step 1

Let T be a linear operator on a finite-dimensional inner product space V.

(a)

Let T be an orthogonal projection.

To prove-

Step 2

Proof

Step 3

The example for which the ine...

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