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Answer: <\/strong>True.<\/div>\n\n
Reasoning:<\/strong> The inclusion \\( \\mathrm{Span}(S)\\subset V \\) is clear, since \\( S\\subset V \\). So let's show that \\( V\\subset \\mathrm{Span}(S) \\). To this end, we want to show that given any \\( p(x)=a+bx+cx^2\\in V \\), we can find \\( c_1,c_2,c_3\\in\\mathbb{R} \\) such that
\n\\[ a+bx+cx^2 = c_1(1+x+x^2)+c_2(5-2x+2x^2)+c_3(-2+3x+x^2). \\] Equating coefficients, we're lead to the system of equations whose augmented matrix is \\[ \\left[\\begin{array}{ccc|c} 1&5&-2 & a\\\\1&-2&3&b\\\\1&2&1&c \\end{array}\\right]. \\] The rank of the coefficient matrix is computed to be \\( 3 \\), which shows that the system is consistent for all \\( a,b,c\\in\\mathbb{R} \\), meaning there are \\( c_1,c_2,c_3 \\) that do what we want. Thus, \\( p(x)\\in\\mathrm{Span}(S) \\). Since \\( p(x)\\in V \\) was arbitrary, it follows that \\( V\\subset\\mathrm{Span}(S) \\).<\/div>\n","tip":"","notChosenFeedback":""},"text":"
True<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
\n
Answer: <\/strong>True.<\/div>\n\n
Reasoning:<\/strong> The inclusion \\( \\mathrm{Span}(S)\\subset V \\) is clear, since \\( S\\subset V \\). So let's show that \\( V\\subset \\mathrm{Span}(S) \\). To this end, we want to show that given any \\( p(x)=a+bx+cx^2\\in V \\), we can find \\( c_1,c_2,c_3\\in\\mathbb{R} \\) such that
\n\\[ a+bx+cx^2 = c_1(1+x+x^2)+c_2(5-2x+2x^2)+c_3(-2+3x+x^2). \\] Equating coefficients, we're lead to the system of equations whose augmented matrix is \\[ \\left[\\begin{array}{ccc|c} 1&5&-2 & a\\\\1&-2&3&b\\\\1&2&1&c \\end{array}\\right]. \\] The rank of the coefficient matrix is computed to be \\( 3 \\), which shows that the system is consistent for all \\( a,b,c\\in\\mathbb{R} \\), meaning there are \\( c_1,c_2,c_3 \\) that do what we want. Thus, \\( p(x)\\in\\mathrm{Span}(S) \\). Since \\( p(x)\\in V \\) was arbitrary, it follows that \\( V\\subset\\mathrm{Span}(S) \\).<\/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=\\mathcal{P}_2(\\mathbb{R}) \\) be the vector space of polynomials of degree \\( \\leq 2 \\) with coefficients in \\( \\mathbb{R} \\), and let \\( S=\\{1 + x + x^2,\\ 5 - 2x + 2x^2,\\ -2 + 3x +x^2\\} \\subset V \\).<\/p>\n\n
True or False: \\( \\mathrm{Span}(S)=V \\).<\/p>\n"}