\(\newcommand{\bbdef}[1]{\expandafter\newcommand% \csname#1\endcsname{\mathbb{#1}}} % \bbdef{C} \bbdef{F} \bbdef{R} \bbdef{Z} \bbdef{Q} \bbdef{K} \bbdef{N} % %%% SCRIPT COMMANDS: \cala=\mathcal{A}, ... \calz=\mathcal{Z} % \newcounter{let} \setcounter{let}{0} % \loop\stepcounter{let} % \expandafter\edef\csname cal\alph{let}\endcsname% % {\noexpand\mathcal{\Alph{let}}} % \ifnum\thelet<26\repeat \renewcommand{\1}{\mathbf{1}} \newcommand{\0}{\mathbf{0}} \newenvironment{amatrix}[1]{% \left(\begin{array}{@{}*{#1}{r}|r@{}} }{% \end{array}\right) } \newenvironment{llista-exercicis}{% \subsection*{Exercicis recomanats} Els exercicis que segueixen són útils per practicar el material presentat. La numeració és la de~\cite{Bret}. \begin{description}}{\end{description}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} \renewcommand{\setminus}{\smallsetminus} \renewcommand{\Rang}{\operatorname{Rang}} \renewcommand{\rref}{\operatorname{rref}} \renewcommand{\rcef}{\operatorname{rcef}} \renewcommand{\Ker}{\operatorname{Ker}} \renewcommand{\Ima}{\operatorname{Im}} \renewcommand{\Id}{\operatorname{Id}} \renewcommand{\Map}{\operatorname{Map}} \renewcommand{\sign}{\operatorname{Sign}} \renewcommand{\refl}{\operatorname{Refl}} \renewcommand{\Tr}{\operatorname{Tr}} \renewcommand{\multalg}{\operatorname{MultAlg}} \renewcommand{\multgeom}{\operatorname{MultGeom}} \renewcommand{\proj}{\operatorname{Proj}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\K}{\mathbb{K}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\F}{\mathbb{F}} \renewcommand{\CC}{\mathbb{C}} \renewcommand{\C}{\mathbb{C}} \renewcommand{\fX}{\mathfrak{X}} \renewcommand{\SL}{\operatorname{SL}} \renewcommand{\GL}{\operatorname{GL}} \renewcommand{\PSL}{\operatorname{PSL}} \renewcommand{\PGL}{\operatorname{PGL}} %Some common abreviations \renewcommand{\lto}{\longrightarrow} \renewcommand{\dfn}{\ensuremath{:=}} \renewcommand{\surjects}{\twoheadrightarrow} \renewcommand{\injects}{\hookrightarrow} \renewcommand{\id}{\operatorname{Id}} \renewcommand{\tns}[1][]{\otimes_{\!#1}} \renewcommand{\mtx}[4]{\left(\begin{matrix}#1\\#3\end{matrix}\right)} \renewcommand{\mat}[1]{\left(\begin{matrix}#1\end{matrix}\right)} \renewcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} \renewcommand{\smtx}[4]{\left(\begin{smallmatrix}#1\\#3\end{smallmatrix}\right)} \renewcommand{\slz}{\operatorname{SL}_2(\bZ)} \renewcommand{\to}{\longrightarrow} \renewcommand{\dlog}{\operatorname{dlog}} % \renewcommand{\Im}{\operatorname{Im}} % \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\abs}[1]{|#1|} \renewcommand{\slsh}[1]{|_{#1}} \renewcommand{\qed}{\blacksquare} \renewcommand{\Irr}{\operatorname{Irr}} \renewcommand{\Aut}{\operatorname{Aut}} \renewcommand{\Gal}{\operatorname{Gal}} \renewcommand{\Mor}{\operatorname{Mor}} \renewcommand{\Hom}{\operatorname{Hom}} % \renewcommand{\implies}{\Longrightarrow} % \renewcommand{\char}{\operatorname{char}} \renewcommand{\car}{\operatorname{char}} \renewcommand{\mcd}{\operatorname{mcd}} \renewcommand{\gcd}{\operatorname{mcd}} \renewcommand{\Eq}{\operatorname{Eq}} \renewcommand{\res}{\operatorname{res}} \renewcommand{\normaleq}{\trianglelefteq} \renewcommand{\disc}{\operatorname{disc}} \renewcommand{\cala}{\mathcal{A}} \renewcommand{\calb}{\mathcal{B}} \renewcommand{\calc}{\mathcal{C}} \renewcommand{\cald}{\mathcal{D}} \renewcommand{\cale}{\mathcal{E}} \renewcommand{\calp}{\mathcal{P}} \renewcommand{\calq}{\mathcal{Q}} \renewcommand{\calr}{\mathcal{R}} \renewcommand{\cals}{\mathcal{S}} \renewcommand{\calt}{\mathcal{T}}\)
Àlgebra Lineal
Introducció
1
Matrius i equacions lineals
1.1
Matrius
1.2
Operacions amb matrius. Matriu invertible
1.3
Transformacions elementals en matrius
1.4
Criteri d’invertibilitat. Rang d’una matriu
1.5
Resolució de sistemes d’equacions lineals
1.6
Exercicis recomanats
2
Espais vectorials i aplicacions lineals
2.1
Matrius com a aplicacions lineals
2.2
Aplicacions lineals i geometria
2.2.1
Homotècies
2.2.2
Projeccions ortogonals
2.2.3
Reflexions
2.2.4
Rotacions
2.2.5
Lliscaments
2.3
Subespais, generadors i bases
2.4
Suma i intersecció de subespais vectorials
2.5
Aplicacions injectives, exhaustives i bijectives
2.6
Coordenades de vectors
2.7
Espais vectorials
2.8
Exercicis recomanats
3
Diagonalització
3.1
Motivació
3.2
Determinants
3.3
Polinomi característic. Valors i vectors propis
3.4
Vectors propis associats a un valor propi
3.5
Interludi: els nombres complexos
3.6
Matrius sobre
\(\R\)
3.7
Exercicis recomanats
4
Ortogonalitat
4.1
Ortogonalitat a
\(\R^n\)
4.2
El teorema de Pitàgores
4.3
Mètode de Gram-Schmidt
4.4
Aplicacions i matrius ortogonals
4.5
Matriu d’una projecció ortogonal en una base ortonormal
4.6
Mínims quadrats
4.6.1
Recta de regressió
4.6.2
Cas general
4.7
Formes bilineals i productes escalars
4.8
Tota matriu simètrica sobre
\(\mathbb{R}\)
diagonalitza
4.9
Descomposició en valors singulars
4.10
Classificació de formes bilineals simètriques sobre
\(\mathbb{R}^n\)
4.11
Exercicis recomanats
Bibliografia
Publicat amb bookdown
Apunts d’Àlgebra Lineal
Bibliografia
1.
Bretscher O (1997) Linear algebra with applications. Prentice Hall Eaglewood Cliffs, NJ
2.
Nart E, Xarles X (2016) Apunts d’
à
lgebra linial. Servei de Publicacions de la Universitat Aut
ò
noma de Barcelona
