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.

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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

See similar textbooks

Similar questions

Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
In Exercises 1-12, determine whether T is a linear transformation. T:FF defined by T(f)=f(x2)
Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x. Prove that T is one-to-one.
Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that if T is invertible, then T∗ is invertible and (T∗)−1 = (T−1)∗.
Let X be a real normed linear space, and let K be a convex set in X, having 0 as an interior point. Let h be the support functional of K and define K° = {x* € X*: h(x*) < 1}. Now for xe X, let p(x) = sup (x, x*). Show that p is equal to the Minkowski functional of K.
Let T be a linear operator on an inner product space V, and supposethat IIT(x)ll = llxll for all x . Prove that Tis one-to-one.