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Let B {b₁,b₂,...,} be an orthogonal subset of Rn. What happens when you apply the Gram- Schmidt algorithm to B (i.e., what set does it produce)? Justify your answer with some computations, and also explain in words why your answer makes sense. = (b) Next of all, let's ask ourselves what happens when the set we start with isn't linearly independent. That's not our primary use case, but what do you think happens? Let's start with a specific example. Let B be the set given below. Show that B is not linearly independent. Apply the Gram-Schmidt algorithm to B and see what you get. Is the resulting set an orthogonal basis for span(B)? If not, is there a natural modification that will turn it into an orthogonal basis for span(B)? 2 3 --(800) B = 1 (c) Try to generalize the specific case above. Make a conjecture about what you think will happen if we apply the Gram-Schmidt algorithm to a general finite subset of R", which may not be linearly independent,

Let B {b₁,b₂,...,} be an orthogonal subset of Rn. What happens when you apply the Gram- Schmidt algorithm to B (i.e., what set does it produce)? Justify your answer with some computations, and also explain in words why your answer makes sense. = (b) Next of all, let's ask ourselves what happens when the set we start with isn't linearly independent. That's not our primary use case, but what do you think happens? Let's start with a specific example. Let B be the set given below. Show that B is not linearly independent. Apply the Gram-Schmidt algorithm to B and see what you get. Is the resulting set an orthogonal basis for span(B)? If not, is there a natural modification that will turn it into an orthogonal basis for span(B)? 2 3 --(800) B = 1 (c) Try to generalize the specific case above. Make a conjecture about what you think will happen if we apply the Gram-Schmidt algorithm to a general finite subset of R", which may not be linearly independent,

Advanced Engineering Mathematics
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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Let B = {b₁,b2, ...‚¯} be an orthogonal subset of Rª. What happens when you apply the Gram- Schmidt algorithm to B (i.e., what set does it produce)? Justify your answer with some computations, and also explain in words why your answer makes sense. (b) Next of all, let's ask ourselves what happens when the set we start with isn't linearly independent. That's not our primary use case, but what do you think happens? Let's start with a specific example. Let B be the set given below. Show that B is not linearly independent. Apply the Gram-Schmidt algorithm to B and see what you get. Is the resulting set an orthogonal basis for span(B)? If not, is there a natural modification that will turn it into an orthogonal basis for span(B)? 2 3 -(0) B = (c) Try to generalize the specific case above. Make a conjecture about what you think will happen if we apply the Gram-Schmidt algorithm to a general finite subset of R", which may not be linearly independent,
Transcribed Image Text:Let B = {b₁,b2, ...‚¯} be an orthogonal subset of Rª. What happens when you apply the Gram- Schmidt algorithm to B (i.e., what set does it produce)? Justify your answer with some computations, and also explain in words why your answer makes sense. (b) Next of all, let's ask ourselves what happens when the set we start with isn't linearly independent. That's not our primary use case, but what do you think happens? Let's start with a specific example. Let B be the set given below. Show that B is not linearly independent. Apply the Gram-Schmidt algorithm to B and see what you get. Is the resulting set an orthogonal basis for span(B)? If not, is there a natural modification that will turn it into an orthogonal basis for span(B)? 2 3 -(0) B = (c) Try to generalize the specific case above. Make a conjecture about what you think will happen if we apply the Gram-Schmidt algorithm to a general finite subset of R", which may not be linearly independent,
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