{"id":"5ecd38b0cfaace694e1be409","name":"Introduction to Linear and Matrix Algebra","summary":"Edfinity is supported by the National Science Foundation","description":"\u003cp dir=\"ltr\"\u003eThis is an online homework companion to Introduction to Linear and Matrix Algebra. It comprises hundreds of algorithmic problems carefully organized into problem sets mapped to textbook sections. Use this course as-is, or customize at any level. You can mix-and-match problems from other catalog courses, add problems from the Edfinity problem repository, or write your own.\u003c/p\u003e\u003cp dir=\"ltr\"\u003e\u003cstrong\u003eHow to use this course\u003c/strong\u003e\u003c/p\u003e\u003col\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eHomework\u003c/strong\u003e\u003cstrong\u003e:\u003c/strong\u003e Assign high quality problems with hints and personalized feedback to develop problem-solving skills.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eTesting\u003c/strong\u003e\u003cstrong\u003e:\u003c/strong\u003e Create summative secure online quizzes and tests in minutes.\u0026nbsp;\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eSupplementary resources:\u003c/strong\u003e Embed videos, class notes, and applets alongside assignments.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eIntervention:\u003c/strong\u003e Use rich analytics to identify and monitor at-risk students for timely intervention.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eAnalytics:\u003c/strong\u003e Drill down into student performance and identify problematic or difficult topics.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eFlipped classroom:\u003c/strong\u003e Assign pre-class assignments. Save precious class time for discussions.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eEmporium classes:\u003c/strong\u003e Use Edfinity for individual/group work for large enrollment sections in labs.\u003c/li\u003e\u003c/ol\u003e","description_more":"","price":29.0,"sort_order":0,"active":true,"individual_license":false,"section_id":"5ecc1886ea6d51000413e1c2","permissive_license":false,"type":"course","year":null,"section_license":true,"taxable":false,"purchasable":true,"related_product_ids":[],"paired_product_id":null,"min_quantity":null,"max_quantity":null,"audience":"educator_audience","site_ids":["55843bd46435650003000000"],"package_price":null,"show_toc":true,"author_roles":[],"author_role_title":"Contributors","institution":null,"tags_array":["math","linear_algebra"],"hidden_tags":[],"meta_description":null,"term_pricing":true,"curricula":null,"curricula_course":null,"highlights":"\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-subscript fa-fw mr-2\"\u003e\u003c/i\u003e \u003cstrong\u003eInteractive, algorithmic problems\u003cbr\u003e\u003c/strong\u003eAlgebraic, graphing, open response; randomized variants, hints, and tips\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-hand-holding-heart fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eAdaptive learning and personalization\u003cbr\u003e\u003c/strong\u003eEach student receives personalized support\u003cbr\u003eto fill learning gaps.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-fw fa-code-branch mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eCorequisite course structures\u003cbr\u003e\u003c/strong\u003eUse pre-built corequisite content,\u003cbr\u003eor create your own.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-hands-helping fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eLMS integration and accessibility\u003cbr\u003e\u003c/strong\u003eConnect to your LMS in minutes. \u003ca class=\"”underline”\" href=\"http://edfinity.com/vpat\" target=\"_blank\"\u003eVPAT here\u003c/a\u003e.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-plug fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eWeBWorK-compatible\u003cbr\u003e\u003c/strong\u003eImport and author WeBWorK problems.\u003c/div\u003e\n","preview_assignment":{"id":"5ecc196dea6d51000413e1ca","name":"1 1 Vectors And Vector Operations","product":{"id":"5ecd38b0cfaace694e1be409","name":"Introduction to Linear and Matrix Algebra"},"master_set":{"id":"5ed5515182041a0004a5d368","problems":{"id":"5ed5515182041a0004a5d37c","proof":false,"graphical_geometric":false,"tags_array":[],"grade_level_from":null,"grade_level_to":null,"skill_ids":["5a14315fac370c0004c81880"],"contest_event_ids":[],"header":null,"solution":null,"part_attributes":[{"id":"5ecc182aea6d51000413d8bd","part_id":"5ecc182aea6d51000413d8bc","description":null,"points":4.0,"automatic_manual_credit":true,"format":"ww","pg":"## DESCRIPTION\n## Solve various vector equations for an unknown vector x in R^2.\n## ENDDESCRIPTION\n\n## DBsubject('Linear algebra')\n## DBchapter('Euclidean spaces')\n## DBsection('Vectors')\n## Date('October 8, 2019')\n## Author('Nathaniel Johnston')\n## Institution('Mount Allison University')\n## TitleText1('Introduction to Linear and Matrix Algebra')\n## EditionText1('1')\n## AuthorText1('Nathaniel Johnston')\n## Section1('1.1: Vectors and Vector Addition')\n## Problem1('')\n## KEYWORDS('vectors','vector addition','linear combinations')\n## http://www.njohnston.ca/publications/introduction-to-linear-and-matrix-algebra/\n\n##############################################################\n#  Initialization\n##############################################################\n\nDOCUMENT();      \n\nloadMacros(\n    \"PGstandard.pl\",\n    \"MathObjects.pl\",\n);\n\nTEXT(beginproblem());\n\n##############################################################\n#  Problem Setup\n##############################################################\n\nContext(\"Vector\");\nContext()-\u003elists-\u003eset(Vector =\u003e {open =\u003e \"(\", close =\u003e \")\"});\nContext()-\u003eparens-\u003eset(\"(\" =\u003e {type =\u003e \"Vector\", close =\u003e \")\"});\n\n$a1 = random(-5,5);\n$b1 = random(-5,5);\n$v1 = Vector([$a1,$b1]);\n    \n$c1 = random(-5,5);\n$d1 = random(-5,5);\n$w1 = Vector([$c1,$d1]);\n    \n$e1 = list_random(-5,-4,-3,-2,2,3,4,5);\n$ans1 = ($w1 - $v1)/$e1;\n\ndo {\n    $a2 = random(-5,5);\n    $b2 = random(-5,5);\n    $v2 = Vector([$a2,$b2]);\n    \n    $c2 = random(-5,5);\n    $d2 = random(-5,5);\n    $w2 = Vector([$c2,$d2]);\n    \n    $e2 = list_random(2,3,4,5);\n} until (($a2 != 0 || $b2 != 0) \u0026\u0026 ($c2 != 0 || $d2 != 0));\n\n$ans2 = ($w2 - $v2)/($e2-1);\n\ndo {\n    $a3 = random(-5,5);\n    $b3 = random(-5,5);\n    $v3 = Vector([$a3,$b3]);\n    \n    $c3 = random(-5,5);\n    $d3 = random(-5,5);\n    $w3 = Vector([$c3,$d3]);\n    \n    $g3 = random(-5,5);\n    $h3 = random(-5,5);\n    $x3 = Vector([$g3,$h3]);\n    \n    $e3 = list_random(-5,-4,-3,-2,2,3,4,5);\n    $f3 = list_random(2,3,4,5);\n} until (($a3 != 0 || $b3 != 0) \u0026\u0026 ($c3 != 0 || $d3 != 0));\n\n$ans3 = ($x3 - $w3 - $e3*$v3) / ($e3*$f3);\n\ndo {\n    $a4 = random(2,6);\n    $b4 = random(2,6);\n} until ($a4 != $b4);\n\n$ans4 = Vector([0,0]);\n\n##############################################################\n#  Text\n##############################################################\n\nContext()-\u003etexStrings;\nBEGIN_TEXT\nSolve the following equations for the vector \\( \\displaystyle \\mathbf{x} \\in \\mathbb{R}^2 \\):\n$BR\n$BR\nIf \\( \\displaystyle $e1 \\mathbf{x} + $v1 = $w1 \\) then \\( \\mathbf{x} = \\) \\{ $ans1-\u003eans_array \\}.\n$BR\nIf \\( \\displaystyle $v2 - \\mathbf{x} = $w2 - $e2 \\mathbf{x} \\) then \\( \\mathbf{x} = \\) \\{ $ans2-\u003eans_array \\}.\n$BR\nIf \\( \\displaystyle $e3\\big($f3 \\mathbf{x} + $v3\\big) + $w3 = $x3 \\) then \\( \\mathbf{x} = \\) \\{ $ans3-\u003eans_array \\}.\n$BR\nIf \\( \\displaystyle $a4(\\mathbf{x} + $a4(\\mathbf{x} + $a4 \\mathbf{x})) = $b4(\\mathbf{x} + $b4(\\mathbf{x} + $b4 \\mathbf{x})) \\) then \\( \\mathbf{x} = \\) \\{ $ans4-\u003eans_array \\}.\nEND_TEXT\nContext()-\u003enormalStrings;\n\n##############################################################\n#  Answers and Solutions\n##############################################################\n\n$showPartialCorrectAnswers = 1;\n\nANS( $ans1-\u003ecmp() );\nANS( $ans2-\u003ecmp() );\nANS( $ans3-\u003ecmp() );\nANS( $ans4-\u003ecmp() );\n\nENDDOCUMENT();","use_equation_editor":null,"webwork_features":[],"part_attributes":[{"id":"5ecc182bea6d51000413d8c7","part_id":"5ecc182bea6d51000413d8bf","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"\u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e(\u003c/SPAN\u003e\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e1.25\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e-1.25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e)\u003c/SPAN\u003e\u003c/span\u003e\u003c/span\u003e","format":"ws","answer_name":"AnSwEr0001","extra":false,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":"\\(\\left(1.25,-1.25\\right)\\)"},{"id":"5ecc182bea6d51000413d8c8","part_id":"5ecc182bea6d51000413d8c0","description":null,"points":1.0,"automatic_manual_credit":true,"answer":null,"format":"ws","answer_name":"MaTrIx_AnSwEr0001_0_1","extra":true,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":null},{"id":"5ecc182bea6d51000413d8c9","part_id":"5ecc182bea6d51000413d8c1","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"\u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e(\u003c/SPAN\u003e\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e2\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e-6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e)\u003c/SPAN\u003e\u003c/span\u003e\u003c/span\u003e","format":"ws","answer_name":"AnSwEr0002","extra":false,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":"\\(\\left(2,-6\\right)\\)"},{"id":"5ecc182bea6d51000413d8ca","part_id":"5ecc182bea6d51000413d8c2","description":null,"points":1.0,"automatic_manual_credit":true,"answer":null,"format":"ws","answer_name":"MaTrIx_AnSwEr0002_0_1","extra":true,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":null},{"id":"5ecc182bea6d51000413d8cb","part_id":"5ecc182bea6d51000413d8c3","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"\u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e(\u003c/SPAN\u003e\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e-0.75\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e2.125\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e)\u003c/SPAN\u003e\u003c/span\u003e\u003c/span\u003e","format":"ws","answer_name":"AnSwEr0003","extra":false,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":"\\(\\left(-0.75,2.125\\right)\\)"},{"id":"5ecc182bea6d51000413d8cc","part_id":"5ecc182bea6d51000413d8c4","description":null,"points":1.0,"automatic_manual_credit":true,"answer":null,"format":"ws","answer_name":"MaTrIx_AnSwEr0003_0_1","extra":true,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":null},{"id":"5ecc182bea6d51000413d8cd","part_id":"5ecc182bea6d51000413d8c5","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"\u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e(\u003c/SPAN\u003e\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e0\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\u003cSPAN STYLE=\"font-family: symbol; margin:0px 2px\"\u003e)\u003c/SPAN\u003e\u003c/span\u003e\u003c/span\u003e","format":"ws","answer_name":"AnSwEr0004","extra":false,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":"\\(\\left(0,0\\right)\\)"},{"id":"5ecc182bea6d51000413d8ce","part_id":"5ecc182bea6d51000413d8c6","description":null,"points":1.0,"automatic_manual_credit":true,"answer":null,"format":"ws","answer_name":"MaTrIx_AnSwEr0004_0_1","extra":true,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":null}],"interactive_html":"\u003cp\u003e\nSolve the following equations for the vector \\(\\displaystyle \\mathbf{x} \\in \\mathbb{R}^2\\):\n\u003cbr\u003e\n\u003cbr\u003e\nIf \\(\\displaystyle -4 \\mathbf{x} + \\left(5,-5\\right) = \\left(0,0\\right)\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"text\" class=\"codeshard\" size=\"5\" name=\"AnSwEr0001\" id=\"AnSwEr0001\" aria-label=\"answer 1 row 1 column 1 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0001\" value=\"\"\u003e\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0001_0_1\" id=\"MaTrIx_AnSwEr0001_0_1\" class=\"codeshard\" aria-label=\"answer 1 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle \\left(3,1\\right) - \\mathbf{x} = \\left(5,-5\\right) - 2 \\mathbf{x}\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"text\" class=\"codeshard\" size=\"5\" name=\"AnSwEr0002\" id=\"AnSwEr0002\" aria-label=\"answer 2 row 1 column 1 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0002\" value=\"\"\u003e\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0002_0_1\" id=\"MaTrIx_AnSwEr0002_0_1\" class=\"codeshard\" aria-label=\"answer 2 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle -4\\big(2 \\mathbf{x} + \\left(2,-5\\right)\\big) + \\left(-1,1\\right) = \\left(-3,4\\right)\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"text\" class=\"codeshard\" size=\"5\" name=\"AnSwEr0003\" id=\"AnSwEr0003\" aria-label=\"answer 3 row 1 column 1 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0003\" value=\"\"\u003e\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0003_0_1\" id=\"MaTrIx_AnSwEr0003_0_1\" class=\"codeshard\" aria-label=\"answer 3 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle 2(\\mathbf{x} + 2(\\mathbf{x} + 2 \\mathbf{x})) = 6(\\mathbf{x} + 6(\\mathbf{x} + 6 \\mathbf{x}))\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"text\" class=\"codeshard\" size=\"5\" name=\"AnSwEr0004\" id=\"AnSwEr0004\" aria-label=\"answer 4 row 1 column 1 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0004\" value=\"\"\u003e\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0004_0_1\" id=\"MaTrIx_AnSwEr0004_0_1\" class=\"codeshard\" aria-label=\"answer 4 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e","static_html":"\u003cp\u003e\nSolve the following equations for the vector \\(\\displaystyle \\mathbf{x} \\in \\mathbb{R}^2\\):\n\u003cbr\u003e\n\u003cbr\u003e\nIf \\(\\displaystyle -4 \\mathbf{x} + \\left(5,-5\\right) = \\left(0,0\\right)\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cspan class=\"ww-blank\" name=\"AnSwEr0001\"\u003e \u003c/span\u003e\n\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0001_0_1\" id=\"MaTrIx_AnSwEr0001_0_1\" class=\"codeshard\" aria-label=\"answer 1 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle \\left(3,1\\right) - \\mathbf{x} = \\left(5,-5\\right) - 2 \\mathbf{x}\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cspan class=\"ww-blank\" name=\"AnSwEr0002\"\u003e \u003c/span\u003e\n\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0002_0_1\" id=\"MaTrIx_AnSwEr0002_0_1\" class=\"codeshard\" aria-label=\"answer 2 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle -4\\big(2 \\mathbf{x} + \\left(2,-5\\right)\\big) + \\left(-1,1\\right) = \\left(-3,4\\right)\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cspan class=\"ww-blank\" name=\"AnSwEr0003\"\u003e \u003c/span\u003e\n\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0003_0_1\" id=\"MaTrIx_AnSwEr0003_0_1\" class=\"codeshard\" aria-label=\"answer 3 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\n\u003cbr\u003e\nIf \\(\\displaystyle 2(\\mathbf{x} + 2(\\mathbf{x} + 2 \\mathbf{x})) = 6(\\mathbf{x} + 6(\\mathbf{x} + 6 \\mathbf{x}))\\) then \\(\\mathbf{x} =\\) \u003cspan class=\"ans_array\" style=\"display:inline-block;vertical-align:.5ex\"\u003e\u003cspan class=\"ans_array_open\" style=\"display:inline-block; vertical-align:middle; margin-right:4px\"\u003e\\(\\left(\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003cspan class=\"ans_array_table\" style=\"display:inline-table; vertical-align:middle\"\u003e\u003cspan style=\"display:table-row\"\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cspan class=\"ww-blank\" name=\"AnSwEr0004\"\u003e \u003c/span\u003e\n\n\u003c/span\u003e\u003cspan class=\"ans_array_sep\" style=\"display:table-cell;vertical-align:middle;padding:0 2px\"\u003e,\u003c/span\u003e\u003cspan class=\"ans_array_cell\" style=\"display:table-cell;vertical-align:middle;padding:4px 0;\"\u003e\u003cinput type=\"TEXT\" size=\"5\" name=\"MaTrIx_AnSwEr0004_0_1\" id=\"MaTrIx_AnSwEr0004_0_1\" class=\"codeshard\" aria-label=\"answer 4 row 1 column 2 \" value=\"\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"ans_array_close\" style=\"display:inline-block; vertical-align:middle; margin-left:4px\"\u003e\\(\\left)\\Rule{0pt}{1.2em}{0pt}\\right.\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e","header_text":{"css":[],"js":[],"inlinejs":[]}}],"numeric_tolerance":null,"partial_credit":null,"max_attempts":null,"allow_practice":null,"instructional":false,"sketchable":true,"print_num_spacing_lines":3,"print_page_break":false,"readonly":{"created_at":"2020-06-01T19:04:49.544Z","updated_at":"2020-06-01T19:04:49.544Z","version":null,"user_id":"58ecdcd48ccf1800120c8ac3","name":"Algorithmic (WebWorK)","skill_ids":["5a14315fac370c0004c81880"],"current_digest":"CGM14vENuMfmSTddF2PzDe4PgsAgnGhBzGGRKLJ2ZFY=","tags":"linear combinations,vector addition,vectors","points":4.0,"type":"webwork","multipart":true,"shallow":true,"static":true,"sources":[],"owned":false,"url":"/problems/5ed5515182041a0004a5d37c","searchable_text":"Solve the following equations for the vector displaystyle mathbf x in mathbb R 2 If displaystyle e1 mathbf x v1 w1 then mathbf x ans1 ans array If displaystyle v2 mathbf x w2 e2 mathbf x then mathbf x ans2 ans array If displaystyle e3big f3 mathbf x v3big w3 x3 then mathbf x ans3 ans array If displaystyle a4 mathbf x a4 mathbf x a4 mathbf x b4 mathbf x b4 mathbf x b4 mathbf x then mathbf x ans4 ans array","abilities":["read"]}},"readonly":{"shallow":true}}},"readonly":{"level_range":"13","created_at":"2020-05-26T15:41:36.123Z","updated_at":"2024-09-01T18:42:19.417Z","tags":"math,linear_algebra","author_ids":[],"licensed":true,"thumbnail":"https://d2b8iahp55539n.cloudfront.net/assets/course-placeholder-3870b2677a589ef8debd801ef338e9403ad9236f1e4562739f9efbd8e456503c.webp","free":false,"user_price":29.0,"term_price":29.0,"displayed_price":29.0,"levels":"13","author_role_title":"Contributors","abilities":[],"section":{"id":"5ecc1886ea6d51000413e1c2","name":"Introduction to Linear and Matrix Algebra"}},"related_products":[],"toc":{"assignments":[{"id":"5ed5513982041a0004a5d2c7","name":"Edfinity Demo"},{"id":"5ecc196dea6d51000413e1ca","name":"1 1 Vectors And Vector Operations"},{"id":"5ecc196eea6d51000413e1cf","name":"1 2 Lengths, Angles, And The Dot Product"},{"id":"5ecc1972ea6d51000413e1db","name":"1 3 Matrices And Matrix Operations"},{"id":"5ecc1977ea6d51000413e1e7","name":"1 4 Linear Transformations"},{"id":"5ecc197eea6d51000413e1f4","name":"1 A Areas, Volumes, And The Cross Product"},{"id":"5ecc1981ea6d51000413e1fe","name":"1 B Paths In Graphs"},{"id":"5ecc1982ea6d51000413e202","name":"2 1 Systems Of Linear Equations"},{"id":"5ecc1988ea6d51000413e214","name":"2 2 Elementary Matrices And Matrix Inverses"},{"id":"5ecc198dea6d51000413e223","name":"2 3 Subspaces, Spans, And Linear Independence"},{"id":"5ecc1991ea6d51000413e234","name":"2 4 Bases And Rank"},{"id":"5ecc1993ea6d51000413e241","name":"2 A Linear Algebra Over Finite Fields"},{"id":"5ecc1993ea6d51000413e244","name":"2 B Linear Programming"},{"id":"5ecc1993ea6d51000413e247","name":"2 C More About The Rank"},{"id":"5ecc1994ea6d51000413e24c","name":"2 D The Lu Decomposition"},{"id":"5ecc1995ea6d51000413e251","name":"3 1 Coordinate Systems"},{"id":"5ecc1997ea6d51000413e25d","name":"3 2 Determinants"},{"id":"5ecc1999ea6d51000413e269","name":"3 3 Eigenvalues And Eigenvectors"},{"id":"5ecc199cea6d51000413e276","name":"3 4 Diagonalization"},{"id":"5ecc199fea6d51000413e280","name":"3 A More About Determinants"},{"id":"5ecc199fea6d51000413e283","name":"3 B Power Iteration"},{"id":"5ecc199fea6d51000413e286","name":"3 C Complex Eigenvalues Of Real Matrices"},{"id":"5ecc19a0ea6d51000413e289","name":"3 D Linear Recurrence Relations"}]}}