{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

True<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"

Answer:<\/strong> False<\/div>\n\n
Reasoning:<\/strong> Positive-definiteness fails. For instance, if \\(\\vec{z} = \\begin{matrix} i \\\\ 0 \\end{bmatrix}\\) then \\(\\vec{z} \\cdot \\vec{z} = i^2+0^2 = -1 \\lt 0\\).<\/div>\n","notChosenFeedback":"
Answer:<\/strong> False<\/div>\n\n
Reasoning:<\/strong> Positive-definiteness fails. For instance, if \\(\\vec{z} = \\begin{matrix} i \\\\ 0 \\end{bmatrix}\\) then \\(\\vec{z} \\cdot \\vec{z} = i^2+0^2 = -1 \\lt 0\\).<\/div>\n"},"text":"
False<\/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":"
True or False: The dot product on \\( \\mathbb{C}^2 \\) defined by \\[ \\begin{bmatrix} z_1\\\\z_2 \\end{bmatrix} \\cdot \\begin{bmatrix} w_1\\\\w_2 \\end{bmatrix} = z_1w_1 + z_2w_2 \\] is an inner product on \\( \\mathbb{C}^2 \\).<\/p>\n\n
Note:<\/strong> Here \\( \\mathbb{C}^2 \\) is being viewed as a complex vector space.<\/p>\n"}