First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "(This question is not randomised, it is the same as Workbook Q 1.30. You can find a randomised version, see link in workbook.)
\nLet \\(A=\\var{A}\\), \\(B=\\var{B}\\), \\(u=\\var{u}\\) and \\(v=\\var{v}\\).
", "advice": "a) To calculate matrix times vector \\(Ax\\), we calculate this row by row: the first entry of the resulting vector is \"row 1 of the matrix times the vector\". In general:
\n\\[\\begin{pmatrix} a_{11} & a_{12} &a_{13} \\\\ a_{21} & a_{22} & a_{23}\\end{pmatrix} \\begin{pmatrix}x_1\\\\x_2\\\\x_3\\end{pmatrix}= \\begin{pmatrix} a_{11}x_1+a_{12}x_2 +a_{13}x_3\\\\ a_{21}x_1+a_{22}x_2 +a_{23}x_3\\end{pmatrix}\\]
\nSo the vector \\(x\\) needs to have as many entries as the matrix has columns, and the resulting vector \\(Ax\\) has as many entries as \\(A\\) has rows.
\n\\(\\var{A}\\var{u}=\\var{unresolvedAu}=\\var{A*u}\\)
\n\\(\\var{A}\\var{v}=\\var{unresolvedAv}=\\var{A*v}\\)
\n\\(\\var{B}\\var{u}=\\var{unresolvedBu}=\\var{B*u}\\)
\n\\(\\var{B}\\var{v}=\\var{unresolvedBv}=\\var{B*v}\\)
\nb) As explained above, the matrix \\(A\\) can take a vector \\(x\\) with \\(\\var{n}\\) entries as input, and gives a vector \\(Ax\\) with \\(\\var{m}\\) entries as output. This is a matrix transformation \\(T_A\\colon { \\mathbb{R}^{\\var{n}} \\to \\mathbb{R}^{\\var{m}}}\\), i.e. \\(n=\\var{n}\\) and \\(m=\\var{m}\\).
\nc) As \\(n=\\var{n}\\), we calculate \\(\\var{A}\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\end{pmatrix} = \\var{latexAx}\\)
", "rulesets": {}, "extensions": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "matrix([0,1,-2],[1,2,3])", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "matrix([0,0,0],[0,0,0])", "description": "", "templateType": "anything"}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(1,-3,-3)", "description": "", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector(-1,3,2)", "description": "", "templateType": "anything"}, "rawunresolvedAu": {"name": "rawunresolvedAu", "group": "Ungrouped variables", "definition": "map(if({A[k][l]<0,'({A[k][l]})','{A[k][l]}')+'\\\\cdot'+if(u[l]<0,'({u[l]})','{u[l]}')+symbols(m,n)[k][l],[k,l],product(0..m-1,0..n-1))", "description": "collects the terms \\(a_{ij}\\cdot v_j\\) for the matrix times vector multiplication, with plus symbols or new line concatenated ready to put into latex code for the calulcation steps of the matrix.
\nAll the \"if\" things are to put brackets round negative numbers. not using simplify because i prefer \\(\\cdot\\) to \\(\\times\\).
\nWithout the need for brackets, it could just be map('{A[k][l]}\\\\cdot{va[l]}'+symbols(2,2)[k][l],[k,l],product(0..1,0..1))
\nThe \"symbols\" is a function giving the correct addition or new line symbols for the size of the matrix that is being used.
", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything"}, "rawunresolvedAv": {"name": "rawunresolvedAv", "group": "Ungrouped variables", "definition": "map(if({A[k][l]<0,'({A[k][l]})','{A[k][l]}')+'\\\\cdot'+if(v[l]<0,'({v[l]})','{v[l]}')+symbols(m,n)[k][l],[k,l],product(0..m-1,0..n-1))", "description": "collects the terms \\(a_{ij}\\cdot v_j\\) for the matrix times vector multiplication, with plus symbols or new line concatenated ready to put into latex code for the calulcation steps of the matrix.
\nAll the \"if\" things are to put brackets round negative numbers. not using simplify because i prefer \\(\\cdot\\) to \\(\\times\\).
\nWithout the need for brackets, it could just be map('{A[k][l]}\\\\cdot{va[l]}'+symbols(2,2)[k][l],[k,l],product(0..1,0..1))
\nThe \"symbols\" is a function giving the correct addition or new line symbols for the size of the matrix that is being used.
", "templateType": "anything"}, "rawunresolvedBu": {"name": "rawunresolvedBu", "group": "Ungrouped variables", "definition": "map(if({B[k][l]<0,'({B[k][l]})','{B[k][l]}')+'\\\\cdot'+if(u[l]<0,'({u[l]})','{u[l]}')+symbols(m,n)[k][l],[k,l],product(0..m-1,0..n-1))", "description": "collects the terms \\(a_{ij}\\cdot v_j\\) for the matrix times vector multiplication, with plus symbols or new line concatenated ready to put into latex code for the calulcation steps of the matrix.
\nAll the \"if\" things are to put brackets round negative numbers. not using simplify because i prefer \\(\\cdot\\) to \\(\\times\\).
\nWithout the need for brackets, it could just be map('{A[k][l]}\\\\cdot{va[l]}'+symbols(2,2)[k][l],[k,l],product(0..1,0..1))
\nThe \"symbols\" is a function giving the correct addition or new line symbols for the size of the matrix that is being used.
", "templateType": "anything"}, "rawunresolvedBv": {"name": "rawunresolvedBv", "group": "Ungrouped variables", "definition": "map(if({B[k][l]<0,'({B[k][l]})','{B[k][l]}')+'\\\\cdot'+if(v[l]<0,'({v[l]})','{v[l]}')+symbols(m,n)[k][l],[k,l],product(0..m-1,0..n-1))", "description": "collects the terms \\(a_{ij}\\cdot v_j\\) for the matrix times vector multiplication, with plus symbols or new line concatenated ready to put into latex code for the calulcation steps of the matrix.
\nAll the \"if\" things are to put brackets round negative numbers. not using simplify because i prefer \\(\\cdot\\) to \\(\\times\\).
\nWithout the need for brackets, it could just be map('{A[k][l]}\\\\cdot{va[l]}'+symbols(2,2)[k][l],[k,l],product(0..1,0..1))
\nThe \"symbols\" is a function giving the correct addition or new line symbols for the size of the matrix that is being used.
", "templateType": "anything"}, "unresolvedAu": {"name": "unresolvedAu", "group": "Ungrouped variables", "definition": "latex('\\\\begin{pmatrix}'+ concatstrings(rawunresolvedAu) +'\\\\end{'+'pmatrix}')", "description": "for the solution, the calculation steps written out unresolved.
", "templateType": "anything"}, "unresolvedAv": {"name": "unresolvedAv", "group": "Ungrouped variables", "definition": "latex('\\\\begin{pmatrix}'+ concatstrings(rawunresolvedAv) +'\\\\end{'+'pmatrix}')", "description": "for the solution, the calculation steps written out unresolved.
", "templateType": "anything"}, "unresolvedBu": {"name": "unresolvedBu", "group": "Ungrouped variables", "definition": "latex('\\\\begin{pmatrix}'+ concatstrings(rawunresolvedBu) +'\\\\end{'+'pmatrix}')", "description": "for the solution, the calculation steps written out unresolved.
", "templateType": "anything"}, "unresolvedBv": {"name": "unresolvedBv", "group": "Ungrouped variables", "definition": "latex('\\\\begin{pmatrix}'+ concatstrings(rawunresolvedBv) +'\\\\end{'+'pmatrix}')", "description": "for the solution, the calculation steps written out unresolved.
", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "map('x_{k}',k,1..n)", "description": "general vector x. ['x_1','x_2','x_3'] As list because vector can only take numbers as entries.
", "templateType": "anything"}, "latexAx": {"name": "latexAx", "group": "Ungrouped variables", "definition": "latex('\\\\begin{pmatrix}'+ concatstrings(halfrawAx) +'\\\\end{'+'pmatrix}')", "description": "Ax as needed for latex string
", "templateType": "anything"}, "rawAx": {"name": "rawAx", "group": "Ungrouped variables", "definition": "map('{A[k][l]}*'+x[{l}]+symbols2(m,n)[k][l],[k,l],product(0..m-1,0..n-1))", "description": "", "templateType": "anything"}, "halfrawAx": {"name": "halfrawAx", "group": "Ungrouped variables", "definition": "map(string(simplify(expression(concatstrings(rawAx[(n*k)..(n*k+n)])),\"all\"))+symbols3(m)[k],k,0..m-1)", "description": "", "templateType": "anything"}, "Ax": {"name": "Ax", "group": "Ungrouped variables", "definition": "map(string(simplify(expression(concatstrings(rawAx[(n*k)..(n*k+n)])),\"all\")),k,0..m-1)", "description": "Ax as list, can be used as expected answer.
", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "u", "v", "m", "n", "rawunresolvedAu", "rawunresolvedAv", "rawunresolvedBu", "rawunresolvedBv", "unresolvedAu", "unresolvedAv", "unresolvedBu", "unresolvedBv", "x", "rawAx", "halfrawAx", "latexAx", "Ax"], "variable_groups": [], "functions": {"concatstrings": {"parameters": [["input", "list"]], "type": "string", "language": "javascript", "definition": "var output = '';\nvar i;\nfor (i = 0; i < input.length; i++) {\n output += input[i];\n} \nreturn output;"}, "symbols": {"parameters": [["m", "number"], ["n", "number"]], "type": "list", "language": "jme", "definition": "repeat(repeat('+',n-1)+['\\\\\\\\'],m-1)+[repeat('+',n-1)+['']]"}, "symbols2": {"parameters": [["m", "number"], ["n", "number"]], "type": "list", "language": "jme", "definition": "repeat(repeat('+',n-1)+[''],m-1)+[repeat('+',n-1)+['']]"}, "symbols3": {"parameters": [["m", "number"]], "type": "list", "language": "jme", "definition": "repeat('\\\\\\\\',m-1)+['']"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Compute:
\n\\(Au = \\) [[0]]
\n\\(Av= \\) [[1]]
\n\\(Bu= \\) [[2]]
\n\\(Bv= \\) [[3]]
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\n\\(n = \\) [[1]] and \\(m= \\) [[0]]
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| \\(T_A(x)= \\left(\\begin{matrix} \\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix} \\right. \\) | \n[[0]] | \n\\(\\left.\\begin{matrix} \\phantom{.}\\\\\\phantom{.}\\\\\\phantom{.}\\end{matrix} \\right) \\) | \n
| [[1]] | \n