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z2~M?2${0h(n_A4QwH#ey=t&AT3)@ef=`;Eb!*S%pEtl9|>cP#lspZ+HEPp6lU1*=< Tl301^p|VhVAt5cJNA&*#Xh4W0 diff --git a/tikz/fundamental_subspaces.tikz b/tikz/fundamental_subspaces.tikz index c89869b..8488b96 100644 --- a/tikz/fundamental_subspaces.tikz +++ b/tikz/fundamental_subspaces.tikz @@ -43,9 +43,9 @@ % \draw[<->, thick] (10,0) -- (0,0) -- (0,10); % Row Space \filldraw \shapeProperties \rotateOne (\pivotOneX, \pivotOneY) rectangle (\pivotOneX + 5, \pivotOneY + 4.1); - \node at \rotateXY{\pivotOneX + 5 / 2}{\pivotOneY + 4.1 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} + \node at \rotateXY{\pivotOneX + 5 / 2}{\pivotOneY + 4.1 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} {\shortstack{ - \scalebox{2}{\(\rowspace{A}\)} \\ + \scalebox{2}{\(\rowspace{A}\)} \\ \scalebox{1.5}{\(\dim{\left( \rowspace{A} \right)} = r\)} }}; % @@ -53,7 +53,7 @@ \filldraw \shapeProperties \rotateOne (\pivotOneX - 5.3, \pivotOneY - 6) rectangle (\pivotOneX, \pivotOneY); \node at \rotateXY{\pivotOneX - 5.3 / 2}{\pivotOneY - 6 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} {\shortstack{ - \scalebox{2}{\(\nullspace{A}\)} \\ + \scalebox{2}{\(\nullspace{A}\)} \\ \scalebox{1.5}{\(\dim{\left( \nullspace{A} \right)} = n - r\)} }}; % @@ -61,14 +61,14 @@ \filldraw \shapeProperties \rotateTwo (\pivotTwoX, \pivotTwoY) rectangle (\pivotTwoX + 4.4, \pivotTwoY + 5.3); \node at \rotateXY{\pivotTwoX + 4.4 / 2}{\pivotTwoY + 5.3 / 2}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {\shortstack{ - \scalebox{2}{\(\columnspace{A}\)} \\ + \scalebox{2}{\(\columnspace{A}\)} \\ \scalebox{1.5}{\(\dim{\left( \columnspace{A} \right)} = r\)} }}; % Left Null Space \filldraw \shapeProperties \rotateTwo (\pivotTwoX - 6.9, \pivotTwoY - 3.9) rectangle (\pivotTwoX, \pivotTwoY); \node at \rotateXY{\pivotTwoX - 6.9 / 2}{\pivotTwoY - 3.9 / 2}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {\shortstack{ - \scalebox{2}{\(\leftnullspace{A}\)} \\ + \scalebox{2}{\(\leftnullspace{A}\)} \\ \scalebox{1.5}{\(\dim{\left( \leftnullspace{A} \right)} = m - r\)} }}; % @@ -82,30 +82,30 @@ % % Zeros \draw [fill] (\pivotOneX, \pivotOneY) circle [radius=0.05]; - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbfup{0}\)}}}] at (\pivotOneX, \pivotOneY) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{0}\)}}}] at (\pivotOneX, \pivotOneY) {}; \draw [fill] (\pivotTwoX, \pivotTwoY) circle [radius=0.05]; - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbfup{0}\)}}}] at (\pivotTwoX, \pivotTwoY) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{0}\)}}}] at (\pivotTwoX, \pivotTwoY) {}; % % Homogeneous vector \draw [fill] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} circle [radius=0.05]; - \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbfit{x}_h\)}}}] at \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} {}; + \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbf{x}_h\)}}}] at \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} {}; % Particular vector \draw [fill] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} circle [radius=0.05]; - \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbfit{x}_p\)}}}] at \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} {}; + \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbf{x}_p\)}}}] at \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} {}; % b vector \draw [fill] \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} circle [radius=0.05]; - \node[label={[label distance=0.25]0:{\scalebox{1.25}{\(\symbfit{b}\)}}}] at \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {}; + \node[label={[label distance=0.25]0:{\scalebox{1.25}{\(\symbf{b}\)}}}] at \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {}; % General vector \draw [fill] (-1, 1.3) circle [radius=0.05]; - \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbfit{x}=\symbfit{x}_p+\symbfit{x}_h\)}}}][below right] at (-1, 1.3) {}; + \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbf{x}=\symbf{x}_p+\symbf{x}_h\)}}}][below right] at (-1, 1.3) {}; % - % Homogeneous Equation + % Homogeneous Equation \draw [thick] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} -- (\pivotTwoX, \pivotTwoY); \draw [->, thick] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} -- ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}); - % Particular Equation + % Particular Equation \draw [thick] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} -- \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}; \draw [->, thick] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} -- ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}); - % General Equation + % General Equation \draw [thick] (-1, 1.3) -- \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}; \draw [->, thick] (-1, 1.3) -- ({(-1 + \rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}, {(1.3 + \rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}); % Dotted Particular Vector @@ -114,12 +114,12 @@ \draw [dashed] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} -- (-1, 1.3); % % Particular Equation Label - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{A}\symbfit{x}_p=\symbfit{b}\)}}}] at ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{A}\symbf{x}_p=\symbf{b}\)}}}] at ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; % General Equation Label - \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbf{A}\symbfit{x}=\symbfit{b}\)}}}] at ({(-1 + \rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}, {(1.3 + \rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}) {}; + \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbf{A}\symbf{x}=\symbf{b}\)}}}] at ({(-1 + \rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}, {(1.3 + \rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}) / 2}) {}; % Homogeneous Equation Label - \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbf{A}\symbfit{x}_h=\symbfup{0}\)}}}] at ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; - % + \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(\symbf{A}\symbf{x}_h=\symbf{0}\)}}}] at ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; + % % Rank definition \node at (0,6.5) {\scalebox{1.75}{\(r=\rank{\left( \symbf{A} \right)}=\rank{\left( \symbf{A}^\top \right)}\)}}; % diff --git a/tikz/matrix_transformation.tikz b/tikz/matrix_transformation.tikz index ee787c6..f4fef69 100644 --- a/tikz/matrix_transformation.tikz +++ b/tikz/matrix_transformation.tikz @@ -48,9 +48,9 @@ % \draw[<->, thick] (10,0) -- (0,0) -- (0,10); % CoImage \filldraw \shapeProperties \rotateOne (\pivotOneX, \pivotOneY) rectangle (\pivotOneX + 5, \pivotOneY + 4.1); - \node at \rotateXY{\pivotOneX + 5 / 2}{\pivotOneY + 4.1 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} + \node at \rotateXY{\pivotOneX + 5 / 2}{\pivotOneY + 4.1 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} {\shortstack{ - \scalebox{2}{\(\vcoim{\left( T \right)}\)} \\ + \scalebox{2}{\(\vcoim{\left( T \right)}\)} \\ \scalebox{1.5}{\(\dim{\left( \vcoim{\left( T \right)} \right)} = r\)} }}; % @@ -58,7 +58,7 @@ \filldraw \shapeProperties \rotateOne (\pivotOneX - 5.3, \pivotOneY - 6) rectangle (\pivotOneX, \pivotOneY); \node at \rotateXY{\pivotOneX - 5.3 / 2}{\pivotOneY - 6 / 2}{\pivotOneX}{\pivotOneY}{\angleOne} {\shortstack{ - \scalebox{2}{\(\vker{\left( T \right)}\)} \\ + \scalebox{2}{\(\vker{\left( T \right)}\)} \\ \scalebox{1.5}{\(\dim{\left( \vker{\left( T \right)} \right)} = \dim{\left( V \right)} - r\)} }}; % @@ -66,14 +66,14 @@ \filldraw \shapeProperties \rotateTwo (\pivotTwoX, \pivotTwoY) rectangle (\pivotTwoX + 4.4, \pivotTwoY + 5.3); \node at \rotateXY{\pivotTwoX + 4.4 / 2}{\pivotTwoY + 5.3 / 2}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {\shortstack{ - \scalebox{2}{\(\vim{\left( T \right)}\)} \\ + \scalebox{2}{\(\vim{\left( T \right)}\)} \\ \scalebox{1.5}{\(\dim{\left( \vim{\left( T \right)} \right)} = r\)} }}; % CoKernel \filldraw \shapeProperties \rotateTwo (\pivotTwoX - 6.9, \pivotTwoY - 3.9) rectangle (\pivotTwoX, \pivotTwoY); \node at \rotateXY{\pivotTwoX - 6.9 / 2}{\pivotTwoY - 3.9 / 2}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {\shortstack{ - \scalebox{2}{\(\vcoker{\left( T \right)}\)} \\ + \scalebox{2}{\(\vcoker{\left( T \right)}\)} \\ \scalebox{1.5}{\(\dim{\left( \vcoker{\left( T \right)} \right)} = \dim{\left( W \right)} - r\)} }}; % @@ -84,40 +84,40 @@ % Number set \node[label={[label distance=40]180:{\scalebox{3}{\(V\)}}}] at (\pivotOneX,\pivotOneY) {}; \node[label={[label distance=40]0:{\scalebox{3}{\(W\)}}}] at (\pivotTwoX,\pivotTwoY) {}; - % + % % Zeros \draw [fill] (\pivotOneX, \pivotOneY) circle [radius=0.05]; - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbfup{0}\)}}}] at (\pivotOneX, \pivotOneY) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{0}\)}}}] at (\pivotOneX, \pivotOneY) {}; \draw [fill] (\pivotTwoX, \pivotTwoY) circle [radius=0.05]; - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbfup{0}\)}}}] at (\pivotTwoX, \pivotTwoY) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(\symbf{0}\)}}}] at (\pivotTwoX, \pivotTwoY) {}; % % Kernel \draw [fill] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} circle [radius=0.05]; - \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbfit{v}\)}}}] at \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} {}; + \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbf{v}\)}}}] at \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} {}; % Vector \draw [fill] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} circle [radius=0.05]; - \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbfit{v}\)}}}] at \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} {}; + \node[label={[label distance=0.25]180:{\scalebox{1.25}{\(\symbf{v}\)}}}] at \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} {}; % Transformed vector \draw [fill] \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} circle [radius=0.05]; - \node[label={[label distance=0.25]0:{\scalebox{1.25}{\(T\left(\symbfit{v}\right)\)}}}] at \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {}; + \node[label={[label distance=0.25]0:{\scalebox{1.25}{\(T\left(\symbf{v}\right)\)}}}] at \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} {}; % - % Kernel Equation + % Kernel Equation \draw [thick] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} -- (\pivotTwoX, \pivotTwoY); \draw [->, thick] \rotateXY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne} -- ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}); - % Transformation Equation + % Transformation Equation \draw [thick] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} -- \rotateXY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo}; \draw [->, thick] \rotateXY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne} -- ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}); - % Transformation of 0 Equation + % Transformation of 0 Equation \draw [thick] (\pivotOneX , \pivotOneY) -- (\pivotTwoX, \pivotTwoY); \draw [->, thick] (\pivotOneX , \pivotOneY) -- ({(\pivotOneX + \pivotTwoX) / 2}, {(\pivotOneY + \pivotTwoY) / 2}); % % Transformation Equation Label - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(T\left(\symbfit{v}\right)=\symbf{A}\symbfit{v}\)}}}] at ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(T\left(\symbf{v}\right)=\symbf{A}\symbf{v}\)}}}] at ({(\rotateX{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateX{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\rotateY{\pivotTwoX + 0.9}{\pivotTwoY + 1.7}{\pivotTwoX}{\pivotTwoY}{\angleTwo} + \rotateY{\pivotOneX + 1.3}{\pivotOneY + 1.1}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; % Transformation of 0 Equation Label - \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(T\left(\symbfup{0}\right)=\symbfup{0}\)}}}] at ({(\pivotOneX + \pivotTwoX) / 2}, {(\pivotOneY + \pivotTwoY) / 2}) {}; + \node[label={[label distance=0.25]90:{\scalebox{1.25}{\(T\left(\symbf{0}\right)=\symbf{0}\)}}}] at ({(\pivotOneX + \pivotTwoX) / 2}, {(\pivotOneY + \pivotTwoY) / 2}) {}; % Kernel Equation Label - \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(T\left(\symbfit{v}\right)=\symbfup{0}\)}}}] at ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; - % + \node[label={[label distance=0.25]325:{\scalebox{1.25}{\(T\left(\symbf{v}\right)=\symbf{0}\)}}}] at ({(\pivotTwoX + \rotateX{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}, {(\pivotTwoY + \rotateY{\pivotOneX - 1.8}{\pivotOneY - 1.5}{\pivotOneX}{\pivotOneY}{\angleOne}) / 2}) {}; + % % Rank definition \node at (0,6.5) {\scalebox{1.75}{\(r=\rank{\left( T \right)}\)}}; % diff --git a/tikz/vector_position.tikz b/tikz/vector_position.tikz index 5cdfda4..7f98f85 100644 --- a/tikz/vector_position.tikz +++ b/tikz/vector_position.tikz @@ -15,16 +15,16 @@ \begin{tikzpicture} \draw [<->] (0,5) -- (0,0) -- (5,0); % Axes % First Vector - \draw [->, dashed, opacity=0.65] (0,0) -- (1,4); + \draw [->, dashed, opacity=0.65] (0,0) -- (1,4); \node [right] at (0.5,2) {\(\overrightarrow{OA}\)}; - \node [above left] at (1,4) {\(\symbfit{a}\)}; + \node [above left] at (1,4) {\(\symbf{a}\)}; % Second Vector \draw [->, dashed, opacity=0.65] (0,0) -- (4,2); \node [below right] at (2,1) {\(\overrightarrow{OB}\)}; - \node [below right] at (4,2) {\(\symbfit{b}\)}; + \node [below right] at (4,2) {\(\symbf{b}\)}; % Displacement Vector \draw [->, thick] (1,4) -- (4,2); - \node [above right] at (2.5,3) {\(\overrightarrow{AB} = \symbfit{b} - \symbfit{a}\)}; + \node [above right] at (2.5,3) {\(\overrightarrow{AB} = \symbf{b} - \symbf{a}\)}; \node [below left] at (0,0) {\(O\)}; \end{tikzpicture} -\end{document} \ No newline at end of file +\end{document} diff --git a/tikz/vector_projection.tikz b/tikz/vector_projection.tikz index 3946540..05b5fa2 100644 --- a/tikz/vector_projection.tikz +++ b/tikz/vector_projection.tikz @@ -20,17 +20,17 @@ \begin{tikzpicture} \draw [<->] (0,5) -- (0,0) -- (5,0); % Axes % a - \draw [->, thin, opacity=0.65] (0,0) -- (4.5,3.9); - \node [below right] at (4.5,3.9) {\(\symbfit{a}\)}; + \draw [->, thin, opacity=0.65] (0,0) -- (4.5,3.9); + \node [below right] at (4.5,3.9) {\(\symbf{a}\)}; % b \draw [->, thick] (0,0) -- (0.8,4); - \node [left] at (0.7,4) {\(\symbfit{b}\)}; + \node [left] at (0.7,4) {\(\symbf{b}\)}; % p \draw [->, thick] (0,0) -- (2.43655,2.11168); - \node [right] at ({2.43655 / 2 + 0.25},{2.11168/2}) {\(\symbfit{p}=\proj_{\symbfit{a}}\symbfit{b}=\symbfit{a}x\)}; + \node [right] at ({2.43655 / 2 + 0.25},{2.11168/2}) {\(\symbf{p}=\proj_{\symbf{a}}\symbf{b}=\symbf{a}x\)}; % e \draw [->, thick, dashed, opacity=0.65] (2.43655,2.11168) -- (0.8,4); - \node [above right] at (1.61827,3.05584) {\(\symbfit{e}=\symbfit{b}-\symbfit{p}\)}; + \node [above right] at (1.61827,3.05584) {\(\symbf{e}=\symbf{b}-\symbf{p}\)}; % Right angle \draw[-, very thin] (2.27282,2.3006) -- (2.08389,2.13686) -- (2.24763,1.94794); \end{tikzpicture}