Problem 3. Let 1 1 2 A = 10 022 12 (a) Find an ordered basis for ran(A) (b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal basis for ran(A) (c) Find a basis for ker(A) (d) Extend the basis you found in (a) to a basis B for R³ (e) Find Pe→B. Let A be as in problem 3. (a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x) for all x € R³. (b) For each of the following b; compute projran (A) (bi) : 51 [2] = 1 18 = 2 ხვ = Ба = 322 (c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a solution to Ax = bi. Use ker(A) and the change of basis matrix you found in problem 3 to find full solution sets to for each i = 1, 2, 3, 4. Axi = bi
Problem 3. Let 1 1 2 A = 10 022 12 (a) Find an ordered basis for ran(A) (b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal basis for ran(A) (c) Find a basis for ker(A) (d) Extend the basis you found in (a) to a basis B for R³ (e) Find Pe→B. Let A be as in problem 3. (a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x) for all x € R³. (b) For each of the following b; compute projran (A) (bi) : 51 [2] = 1 18 = 2 ხვ = Ба = 322 (c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a solution to Ax = bi. Use ker(A) and the change of basis matrix you found in problem 3 to find full solution sets to for each i = 1, 2, 3, 4. Axi = bi
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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Question
Please show work with steps and solution for parts a,b,c with context from problem 3 picture!
Transcribed Image Text:Problem 3. Let
1
1 2
A
= 10
022
12
(a) Find an ordered basis for ran(A)
(b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal
basis for ran(A)
(c) Find a basis for ker(A)
(d) Extend the basis you found in (a) to a basis B for R³
(e) Find Pe→B.
![Let A be as in problem 3.
(a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x)
for all x € R³.
(b) For each of the following b; compute projran (A) (bi) :
51
[2]
= 1
18
= 2
ხვ
=
Ба
=
322
(c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a
solution to
Ax = bi.
Use ker(A) and the change of basis matrix you found in problem 3 to find full solution
sets to
for each i = 1, 2, 3, 4.
Axi
=
bi](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4217e83c-b4b2-47da-9d7a-2d6031448fca%2Fca7b4646-3aea-4bf0-85c0-b73f82f510aa%2Fcj6xwqf_processed.png&w=3840&q=75)
Transcribed Image Text:Let A be as in problem 3.
(a) Compute the matrix of proj ran(A), that is, find the matrix M so that Mã = projran(A)(x)
for all x € R³.
(b) For each of the following b; compute projran (A) (bi) :
51
[2]
= 1
18
= 2
ხვ
=
Ба
=
322
(c) For each 5; from (b), let ỗ¿ = projran(A) (ō;), each ĥ; has Ĝi Є ran(A) and so there is a
solution to
Ax = bi.
Use ker(A) and the change of basis matrix you found in problem 3 to find full solution
sets to
for each i = 1, 2, 3, 4.
Axi
=
bi
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