A calculus of the absurd
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\LWRmarginnote }[1][]{}\)
\(\newcommand {\marginnote }[2][]{\qquad {\small \textrm {#2}}\LWRmarginnote }\)
\(\def \LWRsidenote {1}\)
\(\newcommand {\sidenotemark }[1][\LWRsidenote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\newcommand {\nicefrac }[3][]{\mathinner {{}^{#2}\!/\!_{#3}}}\)
\(\newcommand {\unit }[2][]{#1 \mathinner {#2}}\)
\(\newcommand {\unitfrac }[3][]{#1 \mathinner {{}^{#2}\!/\!_{#3}}}\)
\(\newcommand {\bm }[1]{\boldsymbol {#1}}\)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\Var }{\operatorname {Var}}\)
\(\newcommand {\Expected }{\operatorname {E}}\)
\(\newcommand {\abs }[1]{\left \lvert #1\right \rvert }\)
\(\newcommand {\norm }[1]{\left \lVert #1\right \rVert }\)
\(\newcommand {\rbrackets }[1]{\left (#1\right )}\)
\(\newcommand {\sbrackets }[1]{\left [#1\right ]}\)
\(\newcommand {\cbrackets }[1]{\left \{#1\right \}}\)
\(\newcommand {\RE }{\operatorname {Re}}\)
\(\newcommand {\IM }{\operatorname {IM}}\)
\(\newcommand {\Span }{\operatorname {span}}\)
22.3.2 Linear independence
This is a \(\text {very important}^{TM}\) concept in linear algebra.
-
Definition 22.3.2 Let \(v_1, v_2, ..., v_n\) be some vectors in a vector space \(\textsf {V}\), and let \(a_1, a_2, ..., a_n\) be some scalars in the field \(\mathbb {K}\) (over which this vector space is defined).
We say these vectors are linearly independent if and only if
\(\seteqnumber{0}{22.}{81}\)\begin{equation} a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0 \implies a_1 = a_2 = ... = a_n = 0. \end{equation}
In words, this means “if the only values for all the \(a\)s which satisify \(a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0\) are when all the \(a\)s are zero, then the vectors are linearly independent”.