2d:null 2e:["$","title",null,{"children":"Mais questões de AL | Vinícius Mello | Observable"}] 33:T49b,{ const { Integer, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "CombLinear"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = Random.randInteger(3, 4); const M = transpose(randLI(3, m, 4)); const a1 = Random.randInteger(-3, 3); const a2 = Random.randInteger(-3, 3); const a3 = Random.randInteger(-3, 3); const A = transpose(Matrix([[a1, a2, a3]])); const v = prod(M, A); const q = {}; q.text = raw`Se os vetores $v=${formatVec(v, 1)}$, $v_1=${formatVec( M, 1 )}$, $v_2=${formatVec(M, 2)}$, $v_3=${formatVec(M, 3)}$ são tais que \[v = \alpha_1v_1+\alpha_2v_2+\alpha_3v_3\], calcule $\alpha_1+\alpha_2+\alpha_3$. `; q.answer = a1 + a2 + a3; return q; }); return downloadButton(`${cat}.xml`, xml); }34:T48d,{ const { Integer, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "Coordenada3"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = 3; const M = transpose(randLI(3, m, 4)); const a1 = Random.randInteger(-3, 3); const a2 = Random.randInteger(-3, 3); const a3 = Random.randInteger(-3, 3); const A = transpose(Matrix([[a1, a2, a3]])); const v = prod(M, A); const q = {}; q.text = raw`Com relação à base \[\mathcal{B}=\{${formatVec( M, 1 )}, ${formatVec(M, 2)}, ${formatVec(M, 3)} \},\] $v=${formatVec( v, 1 )}$ tem coordenadas $[v]_\mathcal{B}= (v_1,v_2,v_3) $. Calcule $v_1+2v_2+3v_3$. `; q.answer = a1 + 2 * a2 + 3 * a3; return q; }); return downloadButton(`${cat}.xml`, xml); }35:T5ea,{ const { Integer, Eval, Plus, Times, value, $, latex, toString } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "CombLinearLD"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = Random.randInteger(3, 4); const M = transpose(randLI(3, m, 4)); const a = Random.randInteger(-3, 3); const b = Random.randInteger(-3, 3); for (let i = 1; i <= m; ++i) M[i][2] = Eval( Plus(Times(Integer(a), M[i][1]), Times(Integer(b), M[i][3])) ); const c = Random.randInteger(-3, 3); const d = Random.randInteger(-3, 3); const A = transpose(Matrix([[c, 0, d]])); const v = prod(M, A); const r = latex($`aaaaa+(${a + b}*bbbbb)+ccccc`) .replace('aaaaa', '\\alpha_1') .replace('bbbbb', '\\alpha_2') .replace('ccccc', '\\alpha_3'); const q = {}; q.text = raw`Se os vetores $v=${formatVec(v, 1)}$, $v_1=${formatVec( M, 1 )}$, $v_2=${formatVec(M, 2)}$, $v_3=${formatVec(M, 3)}$ são tais que \[v = \alpha_1v_1+\alpha_2v_2+\alpha_3v_3\], calcule $${r}$. `; q.answer = c + d; return q; }); return downloadButton(`${cat}.xml`, xml); }36:T490,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "DimSoma"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const di = Random.randInteger(2, 4); let M = randLI(di, 4, 4); if (di == 2) { const N = prod(randLI(2, 2, 4), M); M.push(N[1], N[2]); } else if (di == 3) { const a = Random.randInteger(2, 5); const b = -Random.randInteger(2, 5); const N = prod(Matrix([[a, b, 0]]), M); M.push(N[1]); } M = transpose(M); const q = {}; q.text = raw`Se $U=[${formatVec(M, 1)},${formatVec(M, 2)}]$ e $W=[${formatVec(M, 3)},${formatVec( M, 4 )}]$, calcule $\text{dim}(U + W)$. `; q.answer = di; return q; }); return downloadButton(`${cat}.xml`, xml); }37:T4f1,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "Completamento4"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { let M = randLI(4, 4, 4); M = transpose(M); const q = {}; q.text = raw`Escolha o vetor que estende o conjunto \[S=\{${formatVec( M, 1 )},${formatVec(M, 2)},${formatVec( M, 3 )}\}\] para uma base de $\mathbb{R}^4$.`; q.answer = [raw`$ ${formatVec(M, 4)} $`]; while (q.answer.length != 5) { const a = Random.randInteger(1, 2); const b = -Random.randInteger(1, 2); const c = -Random.randInteger(-1, 1); const N = prod(M, transpose(Matrix([[a, b, c, 0]]))); const t = raw`$ ${formatVec(N, 1)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }38:T49d,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "DimIntersecao"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const di = Random.randInteger(2, 4); let M = randLI(di, 4, 4); if (di == 2) { const N = prod(randLI(2, 2, 4), M); M.push(N[1], N[2]); } else if (di == 3) { const a = Random.randInteger(2, 5); const b = -Random.randInteger(2, 5); const N = prod(Matrix([[a, b, 0]]), M); M.push(N[1]); } M = transpose(M); const q = {}; q.text = raw`Se $U=[${formatVec(M, 1)},${formatVec(M, 2)}]$ e $W=[${formatVec(M, 3)},${formatVec( M, 4 )}]$, calcule $\text{dim}(U \cap W)$. `; q.answer = 4 - di; return q; }); return downloadButton(`${cat}.xml`, xml); }39:T592,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, tr, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "MatrizBase"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { let M = randLI(3, 3, 2); let Bl = randLI(3, 3, 3); M = transpose(M); Bl = transpose(Bl); let B = prod(Bl, M); const q = {}; q.text = raw`Dadas as bases \[\mathcal{B}=\{${formatVec(B, 1)}, ${formatVec( B, 2 )}, ${formatVec(B, 3)} \}\] e \[\mathcal{B'}=\{${formatVec(Bl, 1)}, ${formatVec(Bl, 2)}, ${formatVec( Bl, 3 )} \},\] calcule o traço de $M^{\mathcal{B}}_{\mathcal{B'}}$. Dica: Faça $B$ a matriz cujas colunas são os vetores de $\mathcal{B}$ e $B'$ a matriz cujas colunas são os vetores de $\mathcal{B'}$. Então $v=B[v]_\mathcal{B}$ e $v=B'[v]_\mathcal{B'}$, donde \[(B')^{-1}B [v]_\mathcal{B} = [v]_\mathcal{B'},\] ou seja, \[M^{\mathcal{B}}_{\mathcal{B'}} = (B')^{-1}B.\] `; q.answer = value(tr(M)); return q; }); return downloadButton(`${cat}.xml`, xml); }3a:T88b,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "TransformacaoBase"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = Random.randInteger(2, 3); const n = Random.randInteger(2, 3); const M = Matrix(m, n); forEachEntry(M, (i, j) => { M[i][j] = Integer(Random.randInteger(-3, 3)); }); const B = transpose(randLI(n, n, 3)); let b = Matrix(n, 1); forEachEntry(b, (i, j) => { const s = Random.randInteger(0, 1) == 0 ? 1 : -1; b[i][j] = Integer(s * Random.randInteger(2, 3)); }); b = prod(B, b); const MB = prod(M, B); const r = prod(M, b); const q = {}; if (n == 2) { q.text = raw`Seja $T:\mathbb{R}^${n}\rightarrow \mathbb{R}^${m}$ uma transformação linear. Dado que $T${formatVec( B, 1 )}=${formatVec(MB, 1)}$ e $T${formatVec(B, 2)}=${formatVec( MB, 2 )}$, determine $T${formatVec(b, 1)}$. `; } else { q.text = raw`Seja $T:\mathbb{R}^${n}\rightarrow \mathbb{R}^${m}$ uma transformação linear. Dado que $T${formatVec( B, 1 )}=${formatVec(MB, 1)}$, $T${formatVec(B, 2)}=${formatVec( MB, 2 )}$ e $T${formatVec(B, 3)}=${formatVec( MB, 3 )}$, determine $T${formatVec(b, 1)}$. `; } q.answer = [raw`$ ${formatVec(r, 1)} $`]; while (q.answer.length != 5) { let r = Matrix(n, 1); forEachEntry(r, (i, j) => { r[i][j] = Integer(Random.randInteger(-3, 3)); }); r = prod(B, r); r = prod(M, r); const t = raw`$ ${formatVec(r, 1)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }3b:T853,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "MatrizTransformacao"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = Random.randInteger(2, 4); const M = Matrix(4, m); forEachEntry(M, (i, j) => { M[i][j] = Random.randInteger(-3, 3); }); const D = transpose(randLI(m, m, 3)); const A = Matrix(4, 4); const B = Matrix(4, 4); forEachEntry(A, (i, j) => { if (i == j) { A[i][j] = Integer(1); B[i][j] = Integer(1); } else if (i > j) { A[i][j] = Integer(Random.randInteger(-1, 1)); } else { B[i][j] = Integer(Random.randInteger(-1, 1)); } }); const C = prod(A, B); const Ci = adj(C); const R = prod(C, prod(M, D)); let tb = ''; for (let i = 1; i <= m; ++i) tb = tb + `${formatVec(D, i)}` + (i != m ? ',' : ''); const q = {}; q.text = raw`Seja $T:\mathbb{R}^${m}\rightarrow \mathbb{R}^4$ uma transformação linear definida por \[T(v)=${formatMatrix( M )}v,\] onde interpretamos $v$ como uma matriz coluna. Encontre a matriz $[T]_{\mathcal{B}'}^{\mathcal{B}}$ com relação às bases $\mathcal{B}=\{${tb}\}$ e $\mathcal{B}'=\{${formatVec( Ci, 1 )},${formatVec(Ci, 2)},${formatVec(Ci, 3)},${formatVec(Ci, 4)} \}$ `; q.answer = [raw`$ ${formatMatrix(R)} $`]; while (q.answer.length != 5) { let RR = Matrix(4, m); forEachEntry(R, (i, j) => { RR[i][j] = Integer(value(R[i][j]) + Random.randInteger(-4, 4)); }); const t = raw`$ ${formatMatrix(RR)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }3c:T840,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "MatrizTransformacao3"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = Random.randInteger(2, 4); const M = Matrix(3, m); forEachEntry(M, (i, j) => { M[i][j] = Random.randInteger(-3, 3); }); const D = transpose(randLI(m, m, 3)); const A = Matrix(3, 3); const B = Matrix(3, 3); forEachEntry(A, (i, j) => { if (i == j) { A[i][j] = Integer(1); B[i][j] = Integer(1); } else if (i > j) { A[i][j] = Integer(Random.randInteger(-1, 1)); } else { B[i][j] = Integer(Random.randInteger(-1, 1)); } }); const C = prod(A, B); const Ci = adj(C); const R = prod(C, prod(M, D)); let tb = ''; for (let i = 1; i <= m; ++i) tb = tb + `${formatVec(D, i)}` + (i != m ? ',' : ''); const q = {}; q.text = raw`Seja $T:\mathbb{R}^${m}\rightarrow \mathbb{R}^3$ uma transformação linear definida por \[T(v)=${formatMatrix( M )}v,\] onde interpretamos $v$ como uma matriz coluna. Encontre a matriz $[T]_{\mathcal{B}'}^{\mathcal{B}}$ com relação às bases $\mathcal{B}=\{${tb}\}$ e $\mathcal{B}'=\{${formatVec( Ci, 1 )},${formatVec(Ci, 2)},${formatVec(Ci, 3)} \}$ `; q.answer = [raw`$ ${formatMatrix(R)} $`]; while (q.answer.length != 5) { let RR = Matrix(3, m); forEachEntry(R, (i, j) => { RR[i][j] = Integer(value(R[i][j]) + Random.randInteger(-4, 4)); }); const t = raw`$ ${formatMatrix(RR)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }3d:T7be,{ const { Integer, Eval, Plus, Times, subst, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, tr, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "BaseOrtogonal"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const n = 4; let C, ss; do { const S = Matrix(n, n); let d; do { forEachEntry(S, (i, j) => { if (i == j) S[i][j] = Integer(-1); else { if (i > j) S[i][j] = Eval(subst($`-a`, { a: S[j][i] })); else S[i][j] = Integer(Random.randInteger(-3, 4)); } }); d = det(S); } while (value(d) == 0); const R = Matrix(n, n); forEachEntry(R, (i, j) => { if (i == j) R[i][j] = Integer(1); else R[i][j] = S[i][j]; }); C = prod(adj(S), R); ss = value(tr(prod(C, transpose(C)))); } while (ss > 1500); let tb = ''; for (let i = 2; i <= n; ++i) tb = tb + `${formatVec(C, i)}` + (i != n ? ',' : ''); const q = {}; q.text = raw`Qual dos vetores abaixo deve ser acrescentado ao conjunto $S=\{${tb}\}$ para transformá-lo em uma base ortogonal do $\mathbb{R}^${n}$? `; q.answer = [raw`$ ${formatVec(C, 1)} $`]; while (q.answer.length != 5) { let RR = Matrix(n, 1); const c = Random.randInteger(1, n); forEachEntry(RR, (i, j) => { RR[i][j] = Integer( value(C[i][c]) * (Random.randInteger(0, 1) == 0 ? 1 : -1) ); }); const t = raw`$ ${formatVec(RR, 1)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }3e:T41d,{ const { Integer, Eval, Plus, Times, subst, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, tr, adj, randEE } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "DimNucleo"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = 4; const n = 4; const dim = Random.randInteger(2, 4); const T = randEE(m, n, dim, 4); const v = transpose(Matrix([[$`a`, $`b`, $`c`, $`d`]])); const Tv = prod(T, v); const q = {}; q.text = raw`Calcule a dimensão do núcleo da transformação linear $T:\mathbb{R}^4\rightarrow \mathbb{R}^4 $ definida por \[T(a,b,c,d)=${formatVec( Tv, 1 )}.\] `; q.answer = 4 - dim; return q; }); return downloadButton(`${cat}.xml`, xml); }3f:T68c,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "DimAutoespaco"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const dim = Random.randInteger(1, 3); const M = Matrix(4, 4); const a = Random.randInteger(0, 4); const b = Random.randInteger(-4, -1); M[1][1] = Integer(a); M[2][2] = Integer(a); M[3][3] = Integer(a); M[4][4] = Integer(b); for (let i = 1; i <= 3 - dim; ++i) M[i][i + 1] = Integer(1); const A = Matrix(4, 4); const B = Matrix(4, 4); forEachEntry(A, (i, j) => { if (i == j) { A[i][j] = Integer(1); B[i][j] = Integer(1); } else if (i > j) { A[i][j] = Integer(Random.randInteger(-1, 1)); } else { B[i][j] = Integer(Random.randInteger(-1, 1)); } }); const C = prod(A, B); const Ci = adj(C); const R = prod(C, prod(M, Ci)); const q = {}; q.text = raw`Seja $T:\mathbb{R}^4\rightarrow \mathbb{R}^4$ uma transformação linear definida por \[T(v)=${formatMatrix( R )}v,\] onde interpretamos $v$ como uma matriz coluna. Calcule a dimensão do autoespaço $V_\lambda$ associado ao maior autovalor $\lambda$ de $T$. `; q.answer = dim; return q; }); return downloadButton(`${cat}.xml`, xml); }40:T896,{ const { Integer, Eval, Plus, Times, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, adj } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "DiagonalBase"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = 3; const M = Matrix(m, m); let a = Random.randInteger(-3, 3); let b = Random.randInteger(-3, 3); let c = Random.randInteger(-3, 3); [a, b, c] = [a, b, c].sort(); M[1][1] = Integer(a); M[2][2] = Integer(b); M[3][3] = Integer(c); const A = Matrix(m, m); const B = Matrix(m, m); forEachEntry(A, (i, j) => { if (i == j) { A[i][j] = Integer(1); B[i][j] = Integer(1); } else if (i > j) { A[i][j] = Integer(Random.randInteger(-1, 1)); } else { B[i][j] = Integer(Random.randInteger(-1, 1)); } }); const C = prod(A, B); const Ci = adj(C); const R = prod(C, prod(M, Ci)); let tb = ''; for (let i = 1; i <= m; ++i) tb = tb + `${formatVec(C, i)}` + (i != m ? ',' : ''); const q = {}; q.text = raw`Seja $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$ uma transformação linear definida por \[T(v)=${formatMatrix( R )}v,\] onde interpretamos $v$ como uma matriz coluna. Encontre a matriz $[T]_\mathcal{B}$ de $T$ na base $\mathcal{B}=\{${tb}\}$.`; q.answer = [raw`$ ${formatMatrix(M)} $`]; while (q.answer.length != 5) { let MM = Matrix(m, m); let aa = Random.randInteger(-3, 3); let bb = Random.randInteger(-3, 3); let cc = Random.randInteger(-3, 3); [aa, bb, cc] = [aa, bb, cc].sort(); MM[1][1] = Integer(aa); MM[2][2] = Integer(bb); MM[3][3] = Integer(cc); const t = raw`$ ${formatMatrix(MM)} $`; if (q.answer.reduce((a, b) => a && b != t, true)) q.answer.push(t); } return q; }); return downloadButton(`${cat}.xml`, xml); }41:T63b,{ const { Integer, Eval, Plus, Times, subst, value, $, latex } = Guacyra; const { Matrix, randLI, size, transpose, forEachEntry, prod, det, tr, adj, randEE, rankFact, kernel } = LinearAlgebra; const { formatMatrix } = Formatting; const cat = "BaseNucleoImagem"; Random.seed(cat); const formatVec = (M, c) => { const [m, n] = size(M); let s = '('; for (let i = 1; i <= m; ++i) { s = s + latex(M[i][c]); if (i != m) s = s + ', '; } s = s + ')'; return s; }; let xml = MoodleXML(cat, i => { const m = 4; const n = 4; const dim = Random.randInteger(2, 4); const T = randEE(m, n, dim, 4); const v = transpose(Matrix([[$`a`, $`b`, $`c`, $`d`]])); const Tv = prod(T, v); const { F, pivot } = rankFact(T); const K = kernel(F, pivot); console.log(pivot); let tb1 = '['; for (let i = 1; i <= dim; ++i) tb1 = tb1 + `${formatVec(T, pivot[i - 1][1])}` + (i != dim ? ',' : ''); tb1 = tb1 + ']'; let tb2 = '['; for (let i = 1; i <= 4 - dim; ++i) tb2 = tb2 + `${formatVec(K, i)}` + (i != 4 - dim ? 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