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Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option a:<\/strong> Least squares solutions will live in \\( \\mathbb{R}^n \\), not \\( \\mathbb{R}^m \\).<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option a:<\/strong> Least squares solutions will live in \\( \\mathbb{R}^n \\), not \\( \\mathbb{R}^m \\).<\/div>\n","tip":""},"text":"
a. <\/strong>The least squares solution to \\( A\\vec{x}=\\vec{b} \\) is the closest vector to \\( \\vec{b} \\) in \\( \\mathbb{R}^m \\). (Closest with respect to the norm induced by the dot product.)<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option b:<\/strong> A least squares solution is a solution to \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\). If the system \\( A\\vec{x}=\\vec{b} \\) is consistent, that means \\( \\vec{b}\\in\\text{Col}(A) \\) and hence \\( \\text{proj}_{\\text{Col}(A)}(\\vec{b})=\\vec{b} \\). So, the system \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\) is the same as the system \\( A\\vec{x}=\\vec{b} \\).<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option b:<\/strong> A least squares solution is a solution to \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\). If the system \\( A\\vec{x}=\\vec{b} \\) is consistent, that means \\( \\vec{b}\\in\\text{Col}(A) \\) and hence \\( \\text{proj}_{\\text{Col}(A)}(\\vec{b})=\\vec{b} \\). So, the system \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\) is the same as the system \\( A\\vec{x}=\\vec{b} \\).<\/div>\n","tip":""},"text":"
b. <\/strong>A least squares solution to a consistent system \\( A\\vec{x}=\\vec{b} \\) is also a solution to the system.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option c:<\/strong> Least squares solutions are solutions to \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\) and it is perfectly possible for this system to have non-unique solutions. (Give examples!)<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option c:<\/strong> Least squares solutions are solutions to \\( A\\vec{x}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\) and it is perfectly possible for this system to have non-unique solutions. (Give examples!)<\/div>\n","tip":""},"text":"
c.<\/strong> Every system \\( A\\vec{x}=\\vec{b} \\) has a unique least squares solution.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option d:<\/strong> Let \\( \\vec{p}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\). If \\( \\vec{s} \\) is a least squares solution then \\( A\\vec{s}=\\vec{p} \\).<\/div>\n\n
Now, \\( \\vec{b}-\\vec{p} \\in \\text{Col}(A)^\\perp = \\text{Null}(A^T) \\) and so \\( A^T(\\vec{b}-\\vec{p})=\\vec{v}_0 \\) and hence \\( A^T\\vec{b} = A^T\\vec{p} \\). Therefore, \\[ A^T (A\\vec{s}) = A^T(\\vec{p}) = A^T\\vec{b}. \\] Thus \\( \\vec{s} \\) is a solution to \\( A^TA\\vec{x}=A^T\\vec{b} \\).<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are options b and c.<\/div>\n\n
Reasoning for option d:<\/strong> Let \\( \\vec{p}=\\text{proj}_{\\text{Col}(A)}(\\vec{b}) \\). If \\( \\vec{s} \\) is a least squares solution then \\( A\\vec{s}=\\vec{p} \\).<\/div>\n\n
Now, \\( \\vec{b}-\\vec{p} \\in \\text{Col}(A)^\\perp = \\text{Null}(A^T) \\) and so \\( A^T(\\vec{b}-\\vec{p})=\\vec{v}_0 \\) and hence \\( A^T\\vec{b} = A^T\\vec{p} \\). Therefore, \\[ A^T (A\\vec{s}) = A^T(\\vec{p}) = A^T\\vec{b}. \\] Thus \\( \\vec{s} \\) is a solution to \\( A^TA\\vec{x}=A^T\\vec{b} \\).<\/div>\n","tip":""},"text":"
d. <\/strong>A least squares solution to \\( A\\vec{x}=\\vec{b} \\) is a solution to \\( A^TA\\vec{x}=A^T\\vec{b} \\).<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":false,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":false,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","submitAnswerButton":"Submit","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"
Let \\( A\\in M_{m\\times n}(\\mathbb{R}) \\) and \\( \\vec{b}\\in\\mathbb{R}^m \\). Select all true statements.<\/p>\n"}