Answered: 3 6. Work Problem 3 (e ): v1 = 0 |, U2… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 29EQ

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Question

3
6.
Work Problem 3 (-
): v1 =
D2 =
0 2 vectors from
2
R.
(a) [' Determine if the vectors {v1, v2} are linearly
independent
(b) Is it possible to find a vector v3, such that the set
{U1, v2, v3} form a basis for R?

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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.
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Find a standard basis vector that can be added to the set (v1, v2} to produce a basis of R^3. v1 = (1, -1, 0) and v2 = (3, 1, -2) I've tried this, but I found that e1, e2, and e3 form det = 0. Is it right that e1, e2, e3 meet all the requirements?
Let A =[ 4 -1 2 , -1 8 3 ,1 -2 5 ] b=[1 ,-2,3] and initial vector x 0=[0,0,0,0] perform two itearation of the gauss seidel method
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Let u1 = 1 2 3, u2 = 1 −2 1, u3 = 4 1 −2, and y = 1 −3 11(a) Show that {u1, u2, u3} is an orthogonal basis for R3.(b) Express y as a linear combination of u1, u2, and u3.(c) Normalize u1, u2, and u3 to produce an orthonormal basis {v1, v2, v3} for R3
For part b) do we need to check whether the list (x, x, x/2) is also linearly independent? Since a basis of V is a list of vectors in V that is linearly independent and spans V and we know that span = (x, x, x/2) there is also the requirement of linear independence?
Under what conditions on the sacalr 'a' are the vectors (a,1,0),(1,a,1) and (0,1,a) in R\power{3} linearly dependent?
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