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Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option a:<\/strong> This is one of the key uses of the Gram--Schmidt process.<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option a:<\/strong> This is one of the key uses of the Gram--Schmidt process.<\/div>\n","tip":""},"text":"
a. <\/strong>GSP can be used to produce an orthogonal basis for \\( W \\).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option b:<\/strong> GSP takes as input any set of vectors.<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option b:<\/strong> GSP takes as input any set of vectors.<\/div>\n","tip":""},"text":"
b.<\/strong> GSP requires as input an orthogonal basis for \\( W \\).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option c:<\/strong> If the spanning set contains a linear dependency, GSP will produce zero vectors as output. Hence the output will be linearly dependent and therefore not a basis.<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option c:<\/strong> If the spanning set contains a linear dependency, GSP will produce zero vectors as output. Hence the output will be linearly dependent and therefore not a basis.<\/div>\n","tip":""},"text":"
c. <\/strong>Given a spanning set for \\( W \\), GSP will produce an orthogonal basis for \\( W \\).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option d: <\/strong>If GSP is applied to a linearly independent set \\( S \\), it will produce an orthogonal basis for \\( \\mathrm{Span}(S) \\), which is not necessarily equal to \\( W \\) if \\( S \\) does not span \\( W \\).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option d: <\/strong>If GSP is applied to a linearly independent set \\( S \\), it will produce an orthogonal basis for \\( \\mathrm{Span}(S) \\), which is not necessarily equal to \\( W \\) if \\( S \\) does not span \\( W \\).<\/div>\n","tip":""},"text":"
d.<\/strong> Given a linearly independent set in \\( W \\), GSP will produce an orthogonal basis for \\( W \\).<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option e: <\/strong>By the previous point, if \\( S \\) is a basis for \\( W \\), then \\( GSP \\) will produce an orthogonal basis for \\( \\mathrm{Span}(S)=W \\).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are options a and e.<\/div>\n\n
Reasoning for option e: <\/strong>By the previous point, if \\( S \\) is a basis for \\( W \\), then \\( GSP \\) will produce an orthogonal basis for \\( \\mathrm{Span}(S)=W \\).<\/div>\n","tip":""},"text":"
e. <\/strong>Given a basis for \\( W \\), GSP will produce an orthogonal basis for \\( W \\).<\/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 \\( W \\) be a subspace of an inner product space \\( V \\). Select all true statements about the Gram--Schmidt process (GSP).<\/p>\n"}