(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 167269, 5199] NotebookOptionsPosition[ 135967, 4418] NotebookOutlinePosition[ 153098, 4734] CellTagsIndexPosition[ 153016, 4729] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Tema 4: Vectores, matrices y cuestiones de \[CapitalAAcute]lgebra Lineal.\ \>", "Title", CellChangeTimes->{ 3.428129007015625*^9, 3.457844602890625*^9, {3.459144477859375*^9, 3.459144520984375*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["\<\ Definici\[OAcute]n de vectores y matrices \ \>", "Section 1", CellChangeTimes->{ 3.428129142515625*^9, {3.45914474253125*^9, 3.45914478740625*^9}, 3.45914498246875*^9}, TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " permite, por medio de las listas, trabajar de manera muy c\[OAcute]moda \ con vectores y matrices.Para introducir un vector, simplemente introduciremos \ en una lista sencilla los elementos del vector separados por comas .\n\nEl \ comando ", StyleBox["MatrixForm", FontWeight->"Bold"], " nos permite obtener una salida en pantalla similar a la representaci\ \[OAcute]n de matrices y vectores que se hace en la pizarra, pero hay que \ tener cuidado con este comando porque el resultado no es una matriz sino un \ objeto gr\[AAcute]fico con el que posteriormente no se podr\[AAcute] operar \ matricialmente. \n\nPara definir el vector ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["v", "1"]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["v", "2"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["v", "3"], TraditionalForm]]], ") y almacenarlo en la variable ", StyleBox["vectorv", 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4\nDefinir los siguientes vectores: \nw=(1,-5,3), un vector gen\ \[EAcute]rico v de dimensi\[OAcute]n cuatro cuyos elementos sean ", Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]], "None"] }], "Subsection", CellChangeTimes->{{3.4275230702223*^9, 3.427523135093449*^9}, { 3.4275231881054296`*^9, 3.42752318911399*^9}, 3.42812914253125*^9, { 3.457842755359375*^9, 3.457842830796875*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"w", "=", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "5"}], ",", "3"}], "}"}]}]], "Input", CellChangeTimes->{ 3.42812914253125*^9, {3.45784286753125*^9, 3.457842869890625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "5"}], ",", "3"}], "}"}]], "Output", CellChangeTimes->{3.4570669293291087`*^9, 3.45784248*^9, 3.4578428703125*^9, 3.46147764875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "w", "]"}]], "Input", CellChangeTimes->{{3.4275231564241447`*^9, 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RowBox[{"{", RowBox[{ SubscriptBox["v", "1"], ",", SubscriptBox["v", "2"], ",", SubscriptBox["v", "3"], ",", SubscriptBox["v", "4"]}], "}"}]], "Output", CellChangeTimes->{ 3.4570669293916087`*^9, 3.4578424800625*^9, {3.457842856*^9, 3.45784288825*^9}, 3.461477648796875*^9}] }, Open ]], Cell[TextData[{ "En ", StyleBox["Mathematica", FontSlant->"Italic"], " una matriz es una lista de listas, donde cada lista define una fila de la \ matriz. Para introducir una matriz de 2x2 se puede la paleta BasicMathInput o \ BasicTypesetting. Para a\[NTilde]adir filas (ctrl+return), para \ a\[NTilde]adir columnas(ctrl+,). \n\nEl comando Table tambi\[EAcute]n permite \ generar matrices puesto que se pueden poner varios parametros que van \ variando dentro de las opciones del comando, como se puede ver en la ayuda." }], "Text", CellChangeTimes->{{3.427522094475822*^9, 3.427522099624552*^9}, { 3.427522153918236*^9, 3.427522174975996*^9}, {3.427522227092432*^9, 3.427522305968958*^9}, 3.428129142515625*^9, {3.457842335296875*^9, 3.457842392703125*^9}, {3.45914481625*^9, 3.4591448486875*^9}, 3.45914495671875*^9, {3.45914500065625*^9, 3.45914508128125*^9}, { 3.459145364296875*^9, 3.459145367484375*^9}, 3.459145569859375*^9}, TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 2\nDefinir las matrices :\n", Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ { SubscriptBox["a", RowBox[{"1", ",", "1"}]], SubscriptBox["a", RowBox[{"1", ",", "2"}]], SubscriptBox["a", RowBox[{"1", ",", "3"}]]}, 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un vector es una lista de n\[UAcute]meros. Un vector se introduce:\n\ vectorv={", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["v", "1"], ",", SubscriptBox["v", "2"], ",", SubscriptBox["v", "3"]}], "}"}]]], "\n" }], "Text", CellChangeTimes->{{3.427523043048764*^9, 3.427523050580316*^9}, 3.42812914253125*^9}, TextAlignment->Left, TextJustification->1] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Extrayendo elementos de matrices", "Section 1"], Cell[TextData[{ "El elemento de ", StyleBox["Mathematica", FontSlant->"Italic"], " que se utiliza para extraer un elemento de un vector o de una matriz, o \ una fila o una columna de una matriz es el doble corchete: [[ ]].\nEn el \ siguiente ejemplo se utiliza el comando DiagonalMatrix que sirve para crea \ una matriz diagonal que es aquella que tiene todos sus elementos nulos \ excepto los de la diagonal principal. Su sintaxis es la siguiente:\n\t", Cell[BoxData[ FormBox[ RowBox[{"DiagonalMatrix", "[", RowBox[{"{", SubscriptBox["a", "1"]}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["a", "2"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{",", RowBox[{"...", SubscriptBox["a", "n"]}]}], TraditionalForm]]], "}]\nEl resultado ser\[AAcute] una matriz cuadrada de orden n con los \ elementos ", Cell[BoxData[ FormBox[ SubscriptBox["a", "1"], TraditionalForm]]], " , ", Cell[BoxData[ FormBox[ SubscriptBox["a", "2"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{",", RowBox[{"...", SubscriptBox["a", "n"]}]}], TraditionalForm]]], " en la diagonal principal." }], "Text", CellChangeTimes->{{3.45914581959375*^9, 3.45914606240625*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 5\nDefinir la matriz mb = ", Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"2", "4", "6"}, {"3", "6", "9"}, {"5", "10", "15"} }], ")"}], MatrixForm[#]& ]]], ". Extraer:\na) la tercera fila.\nb) la segunda columna\nc) el elemento de \ la 2\.aa fila y 2\.aa columna.\nd) Crear una matriz diagonal con los \ elementos de la diagonal principal de mb" }], "Subsection", CellChangeTimes->{{3.4578429270625*^9, 3.457842952390625*^9}, { 3.457842996625*^9, 3.457843014171875*^9}, {3.45784305121875*^9, 3.457843066375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"mb", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "4", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "6", ",", "9"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "10", ",", "15"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.457843021234375*^9, 3.457843035296875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "mb", "]"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"2", "4", "6"}, {"3", "6", "9"}, {"5", "10", "15"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4570669295166087`*^9, 3.45784248015625*^9, 3.45784303959375*^9, 3.461477649109375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mb", "[", RowBox[{"[", "3", "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"5", ",", "10", ",", "15"}], "}"}]], "Output", CellChangeTimes->{3.4570669295478587`*^9, 3.4578424801875*^9, 3.457843070234375*^9, 3.461477649140625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Transpose", "[", "mb", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]], "Input", CellChangeTimes->{{3.4578430734375*^9, 3.4578430839375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"4", ",", "6", ",", "10"}], "}"}]], "Output", CellChangeTimes->{3.4570669295791087`*^9, 3.45784248021875*^9, 3.45784308471875*^9, 3.46147764915625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mb", "[", RowBox[{"[", RowBox[{"2", ",", "2"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.457843097578125*^9, 3.457843101109375*^9}}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.45784310171875*^9, 3.4614776491875*^9}] }, Open ]], Cell["\<\ Para crear una matriz diagonal se utiliza el comando DiagonalMatrix y se \ introduce la lista de elementos que van en la diagonal principal.\ \>", "Text", CellChangeTimes->{{3.457843105609375*^9, 3.45784315278125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"diag", "=", RowBox[{"DiagonalMatrix", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"mb", "[", RowBox[{"[", RowBox[{"i", ",", "i"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "3"}], "}"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.45784315503125*^9, 3.457843184578125*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "6", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "15"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.457843178515625*^9, 3.45784318509375*^9}, 3.46147764921875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "diag", "]"}]], "Input", CellChangeTimes->{{3.457843186328125*^9, 3.45784319071875*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"2", "0", "0"}, {"0", "6", "0"}, {"0", "0", "15"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.45784319125*^9, 3.461477649234375*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Operaciones con vectores y matrices.", "Section 1", CellChangeTimes->{{3.45914608275*^9, 3.4591460883125*^9}}, FontColor->GrayLevel[0]], Cell[TextData[{ "La multiplicaci\[OAcute]n algebr\[AAcute]ica de vectores y matrices se hace \ con el . o el operador Dot y habr\[AAcute] que tener en cuenta si la \ multiplicaci\[OAcute]n es posible o no seg\[UAcute]n las dimensiones de la \ matriz o vector. Con el operador * se realiza la multiplicaci\[OAcute]n \ elemento a elemento. \n\nEn la siguiente entrada se definen los vectores ", StyleBox["v", FontWeight->"Bold"], " y ", StyleBox["w", FontWeight->"Bold"], ", ", StyleBox["A", FontWeight->"Bold"], ", matriz unidad de orden 4, ", StyleBox["B", FontWeight->"Bold"], ", que no es realmente una matriz sino un objeto de la forma MatrixForm, ", StyleBox["F", FontWeight->"Bold"], " que es una matriz diagonal con los elementos de w en la diagonal principal \ y ", StyleBox["G", FontWeight->"Bold"], " que es una matriz de 4 filas y dos columnas creada con el comando Table." }], "Text", CellChangeTimes->{{3.427523255922716*^9, 3.427523382650858*^9}, { 3.459146206015625*^9, 3.45914632284375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"v", "=", RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}]}], ";", RowBox[{"w", "=", RowBox[{"{", RowBox[{"4", ",", "0", ",", "2", ",", "0"}], "}"}]}], ";", RowBox[{"A", "=", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], ";", RowBox[{"B", "=", RowBox[{"MatrixForm", "[", "A", "]"}]}], ";", RowBox[{"F", "=", RowBox[{"DiagonalMatrix", "[", "w", "]"}]}], ";", RowBox[{"G", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"i", "+", "j"}], ",", RowBox[{"{", RowBox[{"i", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "2"}], "}"}]}], "]"}]}], ";"}]], "Input"], Cell["\<\ El producto elemento a elemento de dos vectores da como resultado otro vector \ tal y como muestra la salida de la siguiente entrada.\ \>", "Text", CellChangeTimes->{{3.459146131*^9, 3.45914618290625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v", "*", "w"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"4", ",", "0", ",", "6", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.4570669296259837`*^9, 3.457842480265625*^9, 3.461477649265625*^9}] }, Open ]], Cell[TextData[{ "En la siguiente entrada se calcula el producto escalar de ", StyleBox["w", FontWeight->"Bold"], " por ", StyleBox["v", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.459146360046875*^9, 3.459146384375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"w", ".", "v"}]], "Input"], Cell[BoxData["10"], "Output", CellChangeTimes->{3.4570669296572337`*^9, 3.45784248028125*^9, 3.46147764934375*^9}] }, Open ]], Cell["\<\ A continuaci\[OAcute]n se calcula el producto vectorial de dos vectores.\ \>", "Text", CellChangeTimes->{{3.4570672784228587`*^9, 3.4570672931416087`*^9}, 3.4570674014697337`*^9, {3.459146420265625*^9, 3.459146439625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cross", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "4"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "3", ",", RowBox[{"-", "2"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4570672968447337`*^9, 3.4570673041572337`*^9}, { 3.4570673517509837`*^9, 3.4570673723759837`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "10"}], ",", "6", ",", "4"}], "}"}]], "Output", CellChangeTimes->{ 3.4570673099697337`*^9, {3.4570673665009837`*^9, 3.4570673727509837`*^9}, 3.45784248034375*^9, 3.461477649390625*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Modulo de un vector", "Section", CellChangeTimes->{{3.459146562015625*^9, 3.45914656615625*^9}}], Cell[TextData[{ "El m\[OAcute]dulo de un vector se puede calcular de varias formas. en la \ siguiente entrada se realiza el c\[AAcute]lculo por medio de la raiz del \ producto escalar de ", StyleBox["v", FontWeight->"Bold"], " por ", StyleBox["v", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.45914647753125*^9, 3.459146530703125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ SqrtBox[ RowBox[{"v", ".", "v"}]]], "Input"], Cell[BoxData[ SqrtBox["30"]], "Output", CellChangeTimes->{3.4570669296884837`*^9, 3.457842480375*^9, 3.4614776494375*^9}] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " proporcional el comando Norm que da el m\[OAcute]dulo del vector asociado \ a la norma eucl\[IAcute]dea." }], "Text", CellChangeTimes->{{3.428129246484375*^9, 3.4281292755*^9}, { 3.45914659159375*^9, 3.45914664390625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Norm", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.428129289*^9, 3.4281292919375*^9}}], Cell[BoxData[ SqrtBox["30"]], "Output", CellChangeTimes->{3.4570669297509837`*^9, 3.4578424804375*^9, 3.45914665559375*^9, 3.4614776495*^9}] }, Open ]], Cell[TextData[{ "Con el comando Normalize se obtiene un vector de norma uno. Se puede \ comprobar que el resultado es el mismo que haciendo ", StyleBox["v", FontWeight->"Bold"], "/Norm[", StyleBox["v", FontWeight->"Bold"], "]" }], "Text", CellChangeTimes->{{3.45914666475*^9, 3.459146709984375*^9}, { 3.459146772359375*^9, 3.459146809234375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Normalize", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.4570674136884837`*^9, 3.4570674170947337`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["1", SqrtBox["30"]], ",", SqrtBox[ FractionBox["2", "15"]], ",", SqrtBox[ FractionBox["3", "10"]], ",", RowBox[{"2", " ", SqrtBox[ FractionBox["2", "15"]]}]}], "}"}]], "Output", CellChangeTimes->{3.4570674179697337`*^9, 3.45784248046875*^9, 3.459146660203125*^9, 3.461477649515625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v", "/", RowBox[{"Norm", "[", "v", "]"}]}]], "Input", CellChangeTimes->{{3.45914683003125*^9, 3.45914685046875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["1", SqrtBox["30"]], ",", SqrtBox[ FractionBox["2", "15"]], ",", SqrtBox[ FractionBox["3", "10"]], ",", RowBox[{"2", " ", SqrtBox[ FractionBox["2", "15"]]}]}], "}"}]], "Output", CellChangeTimes->{3.459146851453125*^9, 3.461477649546875*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Multiplicaci\[OAcute]n de vectores y matrices.\ \>", "Section", CellChangeTimes->{{3.4275235057318172`*^9, 3.4275235366189117`*^9}, 3.427541129229094*^9, {3.4591468804375*^9, 3.4591468886875*^9}}], Cell["\<\ Se puede multiplicar un vector por una matriz de dos formas distintas tal y \ como se ve en las dos entradas siguiente. La tercera muestra la forma \ MatrixForm del resultado.\ \>", "Text", CellChangeTimes->{{3.459146914640625*^9, 3.45914697590625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v", ".", "A"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}]], "Output", CellChangeTimes->{3.4570669297822337`*^9, 3.457842480546875*^9, 3.4614776495625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Dot", "[", RowBox[{"v", ",", "A"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}]], "Output", CellChangeTimes->{3.4570669298134837`*^9, 3.4578424805625*^9, 3.46147764959375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"v", ".", "A"}], "]"}]], "Input", CellChangeTimes->{{3.427523548970626*^9, 3.427523562575128*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ {"1"}, {"2"}, {"3"}, {"4"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4570669298447337`*^9, 3.45784248059375*^9, 3.461477649625*^9}] }, Open ]], Cell[TextData[{ "Haciendo la multiplicaci\[OAcute]n elemento a elemento se obtiene una \ matriz cuya primera fila es la primera fila de ", StyleBox["A", FontWeight->"Bold"], " multiplicada por el primer elemento de ", StyleBox["v", FontWeight->"Bold"], ", etc." }], "Text", CellChangeTimes->{{3.459147031765625*^9, 3.459147100078125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v", "*", "A"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "2", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "3", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "4"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4570669298759837`*^9, 3.457842480625*^9, 3.461477649640625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"v", "*", "A"}], "]"}]], "Input", CellChangeTimes->{{3.42752361321423*^9, 3.427523623495182*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "2", "0", "0"}, {"0", "0", "3", "0"}, {"0", "0", "0", "4"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4570669299541087`*^9, 3.45784248065625*^9, 3.461477649671875*^9}] }, Open ]], Cell["A*v y v*A nos da la misma matriz.", "Text", CellChangeTimes->{{3.427523672096991*^9, 3.4275236737644672`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"A", "*", "v"}], "]"}]], "Input", CellChangeTimes->{{3.42752361321423*^9, 3.427523623495182*^9}, { 3.45914714309375*^9, 3.4591471459375*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "2", "0", "0"}, {"0", "0", "3", "0"}, {"0", "0", "0", "4"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4570669299541087`*^9, 3.45784248065625*^9, 3.45914714778125*^9, 3.4614776496875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}], ".", RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}]}]], "Input", CellChangeTimes->{{3.4570675305791087`*^9, 3.4570675669384837`*^9}}], Cell[BoxData["14"], "Output", CellChangeTimes->{{3.4570675479853587`*^9, 3.4570675682509837`*^9}, 3.4578424806875*^9, 3.46147764971875*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Multiplicaci\[OAcute]n de matrices. ", "Section", CellChangeTimes->{{3.427523679843203*^9, 3.4275237098328857`*^9}, { 3.45914717671875*^9, 3.459147184046875*^9}}], Cell["\<\ Tambi\[EAcute]n se pueden multiplicar elemento a elemento dos matrices, \ siendo necesario que las dos tenga la misma dimensi\[OAcute]n. Debe quedar clara la diferencia entre este tipo de multiplicaci\[OAcute]n y \ la que se define en \[CapitalAAcute]lgebra que se realiza con el punto (.) o \ con el operador Dot y no con el asterisco (*).\ \>", "Text", CellChangeTimes->{{3.45914721015625*^9, 3.45914723253125*^9}, { 3.459147268578125*^9, 3.4591473820625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "*", "F"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "2", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4570669299853587`*^9, 3.45784248071875*^9, 3.461477649765625*^9}] }, Open ]], Cell["\<\ En la siguiente entrada se obtienen las formas MatrixForm de las dos matrices \ y la de su producto elemento a elemento.\ \>", "Text", CellChangeTimes->{{3.4591473984375*^9, 3.45914743359375*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"MatrixForm", "[", "A", "]"}], "\n", RowBox[{"MatrixForm", "[", "F", "]"}], "\n", RowBox[{"MatrixForm", "[", RowBox[{"A", "*", "F"}], "]"}]}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4570669300166087`*^9, 3.45784248075*^9, 3.461477649796875*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"4", "0", "0", "0"}, {"0", "0", "0", 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Entre las m\ \[AAcute]s sencillas:\n\t- la matriz traspuesta: Transpose[A], \n \t- el \ determinante:Det[A] o \n \t- la matriz inversa: Inverse[A]" }], "Text", CellChangeTimes->{{3.4275237893494864`*^9, 3.427523850923853*^9}, { 3.459147714671875*^9, 3.459147738609375*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 6\nDefinir las matrices : a=", Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"1", "2", "3"}, {"2", "4", "7"}, {"5", "7", "10"} }], ")"}], MatrixForm[#]& ]]], " y b=", Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"10", RowBox[{"-", "7"}], "5"}, { RowBox[{"-", "7"}], "4", RowBox[{"-", "2"}]}, { RowBox[{"-", "3"}], "2", RowBox[{"-", "1"}]} }], ")"}], MatrixForm[#]& ]]], ". Calcular:\na) a+b\nb) b-4a\nc) la inversa de a.b\nd) la traspuesta de \ (3a-2b)\ne) el determinante de a." }], "Subsection", CellChangeTimes->{ 3.427523864342804*^9, {3.427541152613456*^9, 3.42754115333346*^9}, { 3.45784345059375*^9, 3.45784353115625*^9}, {3.4578436361875*^9, 3.457843640703125*^9}, 3.45784368790625*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"a", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "4", ",", "7"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "7", ",", "10"}], "}"}]}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"b", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"10", ",", RowBox[{"-", "7"}], ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "7"}], ",", "4", ",", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "3"}]}], "}"}]}], "}"}]}], ";"}]}], "Input", CellChangeTimes->{{3.457843543140625*^9, 3.457843601703125*^9}}], 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Debe \ quedar claro que, por ejemlo la orden matrizb^2 no da como resultado \ matrizb.matrizb, sino otra matriz cuyos elementos son el cuadrado de los \ elementos de matrizb. La sintaxis de este comando es muy sencilla . Para \ elevar la matriz A a la potencia n,\n\tMatrixPower[A,n]" }], "Text", CellChangeTimes->{{3.459147801*^9, 3.459147921046875*^9}}, TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 9\nCalcular Una matriz B cuadrada de orden cuatro cuyos elementos \ sean numeros aleatorios enteros en el intervalo [-9,9]. \nCalcular ", Cell[BoxData[ SuperscriptBox["B", RowBox[{"2", " "}]]]], " y ", Cell[BoxData[ SuperscriptBox["B", "3"]]], ". Obtener otra matriz cuyos elementos sean el cubo de los elementos de B." }], "Subsection", CellChangeTimes->{{3.42754048553726*^9, 3.427540485909977*^9}, 3.427541169007568*^9, 3.4578438955*^9, {3.457844204*^9, 3.45784425334375*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"B", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomInteger", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "9"}], ",", "9"}], "}"}], "]"}], ",", RowBox[{"{", "4", "}"}], ",", RowBox[{"{", "4", "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.457844259078125*^9, 3.45784430515625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "8"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "8", ",", RowBox[{"-", "1"}], ",", RowBox[{"-", "9"}]}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "2", ",", RowBox[{"-", "9"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"8", ",", "5", ",", RowBox[{"-", "8"}], ",", "4"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.457844297484375*^9, 3.457844305984375*^9}, 3.46147765096875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"cuad", "=", RowBox[{"B", ".", "B"}]}], ";", RowBox[{"MatrixForm", "[", "cuad", "]"}]}]], "Input", CellChangeTimes->{{3.4275393424086533`*^9, 3.427539371473034*^9}, { 3.45784433265625*^9, 3.457844333859375*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", "23"}], RowBox[{"-", "12"}], "98", RowBox[{"-", "32"}]}, { RowBox[{"-", "67"}], "9", "41", RowBox[{"-", "106"}]}, { RowBox[{"-", "42"}], "14", RowBox[{"-", "25"}], RowBox[{"-", "34"}]}, { RowBox[{"-", "44"}], "28", RowBox[{"-", "29"}], RowBox[{"-", "61"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.4570669311416087`*^9, 3.45784248228125*^9, {3.457844310578125*^9, 3.4578443349375*^9}, 3.461477650984375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"cub", "=", RowBox[{"MatrixPower", "[", RowBox[{"B", ",", "3"}], "]"}]}], ";", RowBox[{"MatrixForm", "[", "cub", "]"}]}]], "Input", CellChangeTimes->{{3.427539400774762*^9, 3.427539443160121*^9}, 3.4578443393125*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"497", RowBox[{"-", "14"}], RowBox[{"-", "430"}], "522"}, { RowBox[{"-", "190"}], RowBox[{"-", "242"}], "1006", RowBox[{"-", "393"}]}, { RowBox[{"-", "181"}], RowBox[{"-", "24"}], "819", RowBox[{"-", "496"}]}, { RowBox[{"-", "359"}], RowBox[{"-", "51"}], "1073", RowBox[{"-", "758"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.4570669311728587`*^9, 3.457842482328125*^9, {3.45784431284375*^9, 3.45784434021875*^9}, 3.461477651015625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["B", "3"], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.457844322640625*^9, 3.45784434896875*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", "125"}], RowBox[{"-", "8"}], RowBox[{"-", "512"}], "8"}, {"64", "512", RowBox[{"-", "1"}], RowBox[{"-", "729"}]}, {"343", "8", RowBox[{"-", "729"}], "216"}, {"512", "125", RowBox[{"-", "512"}], "64"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.45784432528125*^9, 3.45784434946875*^9}, 3.461477651046875*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Otras Cuestiones de Algebra Lineal", "Section 1", CellChangeTimes->{{3.4591447391875*^9, 3.459144739703125*^9}, { 3.459147988671875*^9, 3.459147990734375*^9}}], Cell["\<\ A una matriz A se le pueden asociar dos espacios vectoriales. El generado por \ los vectores definidos por sus filas, fil (A), y el espacio vectorial \ generado por sus columnas, col (A). La dimensi\[OAcute]n de fil (A) o de col (A) es el rango de la matriz. El comando MatrixRank da el rango de la matriz. Otro espacio vectorial importante es el subespacio nulo o kernel que es el \ generado por el conjunto de soluciones de : A.x = 0. Su dimensi\[OAcute]n se \ denomina nulidad de A. El rango de A es igual al n\[UAcute]mero de filas no nulas en la forma \ escalonada por filas de A y la nulidad de A es igual al numero de filas de \ ceros en dicha forma. Si A es una matriz cuadrada, la suma del rango de A \ mas la nulidad de A es igual al n\[UAcute]mero de filas o de columnas de A. \ El comando RowReduce[A] da la forma escalonada reducida por filas de A. Las \ filas no nulas forman una base del subespacio vectorial generado por las \ filas de A. El comando NullSpace[A] da una lista con los vectores que forman el \ subespacio nulo. \ \>", "Text", CellChangeTimes->{{3.459148025875*^9, 3.459148229421875*^9}, { 3.4591482841875*^9, 3.459148299640625*^9}, {3.459148898171875*^9, 3.45914896678125*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 10\nSea A = ", Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"4", "1", "1", "1"}, {"1", "3", RowBox[{"-", "1"}], "1"}, {"1", RowBox[{"-", "1"}], "2", "0"}, {"3", "2", RowBox[{"-", "1"}], "1"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], CellChangeTimes->{3.4592369075625*^9}], "\nObtener:\n\t- El rango de A\n\t- Su forma escalonada por filas\n\t- Una \ base del subespacio nulo de A\n" }], "Subsection", CellChangeTimes->{ 3.4275258593759737`*^9, 3.4275412652916937`*^9, 3.457932060171875*^9, { 3.45914823859375*^9, 3.45914823915625*^9}, {3.459236844453125*^9, 3.459236916375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"A", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "1", ",", "1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "3", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}], ",", "2", ",", "0"}], "}"}], ",", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "1", ",", "1", ",", "1"}], "}"}], "-", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}], ",", "2", ",", "0"}], "}"}]}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{3.459148254578125*^9, 3.4592368379375*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixRank", "[", "A", "]"}]], "Input", CellChangeTimes->{{3.459236900625*^9, 3.459236923328125*^9}}], Cell[BoxData["3"], "Output", 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12\nHallar una base del subespacio nulo de la matriz \nH = ", Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"1", "2", RowBox[{"-", "2"}], "1"}, { RowBox[{"-", "3"}], "4", "0", "1"}, {"0", "2", "1", "0"} }], ")"}], MatrixForm[#]& ]]] }], "Subsection", CellChangeTimes->{{3.427526080386713*^9, 3.427526080806538*^9}, 3.427541282915275*^9, {3.457932069421875*^9, 3.457932069640625*^9}, { 3.4591489940625*^9, 3.45914901853125*^9}, {3.459237787046875*^9, 3.4592378098125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"H", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", RowBox[{"-", "2"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "3"}], ",", "4", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "2", ",", "1", ",", "0"}], "}"}]}], "}"}]}], ";", RowBox[{"NullSpace", "[", "H", "]"}]}]], "Input", CellChangeTimes->{{3.45914900075*^9, 3.45914902678125*^9}, { 3.4592377328125*^9, 3.45923776709375*^9}}], Cell[BoxData[ RowBox[{"{", 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Los comandos Orthogonalize, y Projection \ nos permiten acceder a operaciones algebr\[AAcute]icas en espacios \ vectoriales como ortonormalizar una base vectorial o realizar la proyecci\ \[OAcute]n de un vector en un subespacio vectorial. Si ", Cell[BoxData[ FormBox[ RowBox[{"(", StyleBox[ SubscriptBox["v", "1"], FontWeight->"Bold"]}], TraditionalForm]]], " , ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["v", "2"], FontWeight->"Bold"], TraditionalForm]]], " ,...) es la base del subespacio vectorial la sintaxis del comando \ Orthogonalize es la siguiente:\n\t", Cell[BoxData[ FormBox[ RowBox[{"Orthogonalize", "[", RowBox[{"{", SubscriptBox["v", "1"]}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["v", "2"], TraditionalForm]]], ", ...}]\nPara proyectar el vector ", StyleBox["u", FontWeight->"Bold"], " sobre el vector ", StyleBox["v", FontWeight->"Bold"], ", la sintaxis del comando Projection es:\n\tProjection[u,v]" }], "Text", CellChangeTimes->{{3.427526194477231*^9, 3.427526244823226*^9}, { 3.427526281992467*^9, 3.4275263678422403`*^9}, {3.459149392625*^9, 3.459149431625*^9}, {3.45914946715625*^9, 3.459149552609375*^9}, { 3.45914962846875*^9, 3.4591496735625*^9}, {3.45915000878125*^9, 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