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Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option a:<\/strong> Counterexample: \\(\\mathbb{R}^2\\) with dot product, \\(\\vec{x}=\\begin{bmatrix} 1\\\\0 \\end{bmatrix}\\), \\(\\vec{y} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\\).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option a:<\/strong> Counterexample: \\(\\mathbb{R}^2\\) with dot product, \\(\\vec{x}=\\begin{bmatrix} 1\\\\0 \\end{bmatrix}\\), \\(\\vec{y} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\\).<\/div>\n"},"text":"
a. <\/strong>If \\( \\langle\\vec{x}, \\vec{y} \\rangle=0 \\) then either \\( \\vec{x}=\\vec{v}_0 \\) or \\( \\vec{y}=\\vec{v}_0 \\).<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option b: <\/strong>True by definition of an inner product (positive-definiteness).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option b: <\/strong>True by definition of an inner product (positive-definiteness).<\/div>\n"},"text":"
b. <\/strong>If \\( \\langle\\vec{x}, \\vec{x} \\rangle=0 \\) then \\( \\vec{x}=\\vec{v}_0 \\).<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option c: <\/strong>True by definition of an inner product (linearity in each argument: \\(\\langle 2\\vec{x}, \\vec{y} \\rangle = 2 \\langle \\vec{x}, \\vec{y} \\rangle = \\langle \\vec{x}, 2\\vec{y} \\rangle\\)).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option c: <\/strong>True by definition of an inner product (linearity in each argument: \\(\\langle 2\\vec{x}, \\vec{y} \\rangle = 2 \\langle \\vec{x}, \\vec{y} \\rangle = \\langle \\vec{x}, 2\\vec{y} \\rangle\\)).<\/div>\n"},"text":"
c. <\/strong>\\( \\langle 2\\vec{x}, \\vec{y} \\rangle= \\langle \\vec{x}, 2\\vec{y} \\rangle \\).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option d: <\/strong>Counterexample: \\(\\mathbb{R}^2\\) with dot product, \\(\\vec{x}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}\\), \\( \\vec{y} = \\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix}\\).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> The correct choices are b and c.<\/div>\n\n
Reasoning for option d: <\/strong>Counterexample: \\(\\mathbb{R}^2\\) with dot product, \\(\\vec{x}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}\\), \\( \\vec{y} = \\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix}\\).<\/div>\n"},"text":"
d. <\/strong>\\( \\langle \\vec{x}, \\vec{y} \\rangle \\) must be non-negative.<\/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 \\( \\langle\\,,\\rangle \\) be an inner product on a real vector space \\( V \\), and let \\( \\vec{x},\\vec{y}\\in V \\). Select all true statements.<\/p>\n"}