matrix analysis practice question,Please write the simplest answer process,thanks. 5. Let U: R → R be linear. Show that the following statements areequivalent:(a) U is an orthogonal matrix.(b) If {x¹,x²,...,x"} is an orthonormal basis of R", then the image set{U (x¹), U(x²),…..‚U(r)} is an orthonormal basis of R.

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Answered: 5. Let U: R →R be linear. Show that the…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
In Exercises 7-10, show that the given vectors form an orthogonal basis for2or3. Then use Theorem 5.2 to express was a linear combination of these basis vectors. Give the coordinate vector[w] ofwwith respect to the basis ={v1,v2}of 2or =v1,v2,v3 of3. v1=[111],v2=[110],v3=[112];w=[123]
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

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5. Let U: R →R be linear. Show that the following statements are equivalent: (a) U is an orthogonal matrix. (b) If {x¹,x²,...,x^} is an orthonormal basis of R", then the image set {U (x¹), U(x²),…..‚U(r)} is an orthonormal basis of R.