(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 53547, 1133] NotebookOptionsPosition[ 51779, 1066] NotebookOutlinePosition[ 52142, 1082] CellTagsIndexPosition[ 52099, 1079] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Least squares solutions of linear systems and orthogonal projections\ \>", "Section", CellChangeTimes->{{3.625568347222041*^9, 3.625568382004538*^9}}], Cell[CellGroupData[{ Cell["\<\ Let A be an m x n matrix and V = Col(A) be the associated subspace of R^n. \ For instance:\ \>", "Subsubsection", CellChangeTimes->{{3.625568392836306*^9, 3.625568444035939*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"m", "=", "3"}], "\[IndentingNewLine]", RowBox[{"n", "=", "2"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", RowBox[{"A", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "1", ",", RowBox[{"-", "1"}]}], "}"}]}], "}"}], "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.624355479294818*^9, 3.624355525906296*^9}, { 3.62556827720107*^9, 3.6255683166081333`*^9}, {3.625568456229638*^9, 3.625568466750351*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.6255760390238166`*^9, 3.6255779420094204`*^9, 3.625578092952118*^9}], Cell[BoxData["2"], "Output", CellChangeTimes->{3.6255760390238166`*^9, 3.6255779420094204`*^9, 3.625578092967717*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"}, {"2", RowBox[{"-", "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->{3.6255760390238166`*^9, 3.6255779420094204`*^9, 3.625578092967717*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "If b is a vector in R^n, then the system Ax = b has a solution if and only \ if b in Col(A). 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