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

Answer: <\/strong>False.<\/div>\n\n
Reasoning:<\/strong> The quick way to see this is to note that \\( \\dim M_{2\\times3}(\\mathbb{F})=6 \\), so \\( M_{2\\times3}(\\mathbb{F}) \\) cannot be spanned by a set containing fewer than 6 vectors. Alternatively, we can see by inspection that \\( \\begin{bmatrix} 0&0&0\\\\0&0&1 \\end{bmatrix}\\notin\\mathrm{Span}(S) \\), since every linear combination of the matrices in \\( S \\) will have a 0 in the \\( (2,3) \\) entry.<\/div>\n","tip":"","notChosenFeedback":""},"text":"
True<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"
\n
Answer: <\/strong>False.<\/div>\n\n
Reasoning:<\/strong> The quick way to see this is to note that \\( \\dim M_{2\\times3}(\\mathbb{F})=6 \\), so \\( M_{2\\times3}(\\mathbb{F}) \\) cannot be spanned by a set containing fewer than 6 vectors. Alternatively, we can see by inspection that \\( \\begin{bmatrix} 0&0&0\\\\0&0&1 \\end{bmatrix}\\notin\\mathrm{Span}(S) \\), since every linear combination of the matrices in \\( S \\) will have a 0 in the \\( (2,3) \\) entry.<\/div>\n<\/div>\n","tip":"","notChosenFeedback":""},"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":"
Let \\( V=M_{2\\times3}(\\mathbb{F}) \\) be the vector space of \\( 2\\times3 \\) matrices with entries in \\( \\mathbb{F} \\), and let \\( S=\\left\\{\\begin{bmatrix} 1 & 1 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix}, \\begin{bmatrix} 2 & 2 & 2 \\\\ 0 & 0 & 0 \\end{bmatrix}\\right\\} \\).<\/p>\n\n
True or False: \\( \\mathrm{Span}(S)=V \\).<\/p>\n"}