Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V(hence dim V = 2).Consider u, uz € V, and define the linear map T : V → V byT(e1) = u1T(e2) = u2.a) Show that u,, uz is a basis of V if and only if|(u1, u2)| < ||us|||42||-b) Show that T is one-to-one if and only if T is onto (in which case it is an isomor-phism).c) Show that T is an isomorphism if and only if |(u), u2)| < ||u,|| ||12||d) Show that ||T(u)|| = ||u|| for all u E V if and only if ||u |ll=||u2|| = 1 and (u1, U2) = 0.

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Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V (hence dim V = 2). Consider u, uz € V, and define the linear map T : V →V by T(e1) = u1 T(e2) = u2. a) Show that u,, uz is a basis of V if and only if |(u1, u2)| < ||u|||uz||. b) Show that T is one-to-one if and only if T is onto (in which case it is an isomor- phism). c) Show that T is an isomorphism if and only if (u, u2) < ||u, || |12|- d) Show that ||T(u)|| = ||u|| for all u E V if and only if || 1||= ||u2|| =1 and (u1, u2) = 0.

Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V (hence dim V = 2). Consider u, uz € V, and define the linear map T : V →V by T(e1) = u1 T(e2) = u2. a) Show that u,, uz is a basis of V if and only if |(u1, u2)| < ||u|||uz||. b) Show that T is one-to-one if and only if T is onto (in which case it is an isomor- phism). c) Show that T is an isomorphism if and only if (u, u2) < ||u, || |12|- d) Show that ||T(u)|| = ||u|| for all u E V if and only if || 1||= ||u2|| =1 and (u1, u2) = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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