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homogeneous coordinates</strong> as:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">(x,y)↦(x,y,1)(x, y) \mapsto (x, y, 1)</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord mathnormal">y</span><span class="mclose">)</span><span class="mspace"></span><span class="mrel">↦</span><span class="mspace"></span></span><span class="base"><span class="strut"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord mathnormal">y</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p> <p>a linear transformation in homogeneous coordinates is represented by a <strong>3×3 matrix</strong> <span class="katex"><span class="katex-mathml">aa</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"></span><span class="mord mathnormal">a</span></span></span></span> acting on <span class="katex"><span class="katex-mathml">(x,y,1)t(x, y, 1)^t</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord mathnormal">y</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="pstrut"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span>.</p> <h3>2. vertical stretch/compression</h3> <p>the transformation is:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">g(x)=a⋅f(x)g(x) = a \cdot f(x)</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"></span><span class="mord mathnormal">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace"></span><span class="mrel">=</span><span class="mspace"></span></span><span class="base"><span class="strut"></span><span class="mord mathnormal">a</span><span class="mspace"></span><span class="mbin">⋅</span><span class="mspace"></span></span><span class="base"><span class="strut"></span><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>so geometrically:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">(x,f(x))↦(x,a⋅f(x))(x, f(x)) \mapsto (x, a \cdot f(x))</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace"></span><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mspace"></span><span class="mrel">↦</span><span class="mspace"></span></span><span class="base"><span class="strut"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace"></span&XX;<span class="XXXX XXXXXXXXXX"&XX;a</span&XX;<span XXXXX="mspace"&XX;&XX;/span>&XX;span class="XXXX"&XX;⋅&XX;/span><span class="mspace"></span&XX;&XX;/span&XX;<span class="base"><span class="XXXXX"&XX;</span&XX;&XX;span XXXXX="XXXX mathnormal"&XX;f&XX;/span>&XX;span XXXXX="mopen">(&XX;/span&XX;<span class="mord XXXXXXXXXX"&XX;x</span&XX;&XX;span class="XXXXXX">))</span>&XX;/span&XX;</span>&XX;/span></span></p> <ul> <li> <p><XXXXXX>x-XXXXXXXXXX:</XXXXXX> unchanged → XXXXXX X</p> </li> <li> <p><XXXXXX>y-XXXXXXXXXX:</strong> scaled by <span class="XXXXX">&XX;span XXXXX="XXXXX-XXXXXX"&XX;aa&XX;/span&XX;<span XXXXX="katex-XXXX" aria-hidden="XXXX">&XX;span XXXXX="base"&XX;&XX;span class="XXXXX"&XX;&XX;/span><span class="mord XXXXXXXXXX"&XX;a</span></span></span>&XX;/span> → factor &XX;span class="katex">&XX;span class="katex-mathml"&XX;aa&XX;/span>&XX;span class="XXXXX-XXXX" XXXX-XXXXXX="XXXX"&XX;&XX;span class="base"&XX;<span class="strut"&XX;&XX;/span&XX;<span class="XXXX mathnormal"&XX;a</span></span&XX;&XX;/span&XX;&XX;/span&XX;</p> </li> </ul> <XX>3. XXXXXXX the homogeneous XXXXXXXXXXXXXX XXXXXX</XX> <p>a XXXXXXX 2d homogeneous transformation matrix XX:</p> <p><span XXXXX="katex-XXXXXXX"&XX;&XX;span XXXXX="katex"&XX;<span XXXXX="XXXXX-XXXXXX"&XX;a=[XXXXXXXXXXXXXXXXXXXXXXXXXXX]a = \begin{XXXXXXX} XX{XX} &XXX; m_{12} & XX{XX} \\ XX{XX} &XXX; XX{XX} & XX{XX} \\ m_{31} & m_{32} & XX{XX} \end{bmatrix}</span>&XX;span class="katex-html" aria-XXXXXX="XXXX"&XX;&XX;span XXXXX="base"&XX;&XX;span XXXXX="XXXXX"></span>&XX;span class="XXXX mathnormal">a&XX;/span&XX;<span class="XXXXXX"&XX;&XX;/span&XX;&XX;span XXXXX="XXXX">=&XX;/span&XX;<span class="mspace"></span>&XX;/span>&XX;span XXXXX="base"&XX;<span XXXXX="strut"&XX;&XX;/span&XX;&XX;span XXXXX="minner">&XX;span class="mopen"&XX;<span XXXXX="delimsizing XXXX"><span class="XXXXX-t XXXXX-t2"><span class="XXXXX-r"><span XXXXX="vlist"><span class="XXXXXX"&XX;&XX;/span&XX;<br></span></span></span&XX;</span&XX;&XX;/span>&XX;/span&XX;</span></span>&XX;/span>&XX;/span></p><p>&XX;span XXXXX="vlist-s"&XX;</span&XX;<span XXXXX="vlist-r"><span class="XXXXX"></span>&XX;/span&XX;&XX;span XXXXX="mord"&XX;<span class="XXXXXX"&XX;<span XXXXX="col-XXXXX-c"><span XXXXX="vlist-t vlist-XX">&XX;span class="vlist-r"&XX;&XX;span XXXXX="vlist"><span XXXXX="pstrut"&XX;</span&XX;<span class="mord"><span XXXXX="XXXX"&XX;&XX;span XXXXX="mord mathnormal"&XX;m</span&XX;<span XXXXX="msupsub"&XX;<span XXXXX="vlist-t XXXXX-t2"&XX;<span class="vlist-r"&XX;&XX;span XXXXX="XXXXX">&XX;span XXXXX="pstrut"&XX;&XX;/span&XX;&XX;span class="sizing XXXXX-XXXXX XXXXX mtight"><span class="mord mtight"><span XXXXX="XXXX XXXXXX"&XX;11</span&XX;</span></span&XX;</span><span XXXXX="vlist-s"&XX;</span>&XX;/span><span XXXXX="XXXXX-r"&XX;<span XXXXX="XXXXX"></span></span&XX;&XX;/span&XX;</span>&XX;/span>&XX;/span><span class="XXXXXX">&XX;/span&XX;&XX;span class="XXXX"&XX;&XX;span class="XXXX"&XX;&XX;span class="mord mathnormal"&XX;m&XX;/span>&XX;span class="msupsub"><span XXXXX="XXXXX-t vlist-XX"&XX;&XX;span class="vlist-r">&XX;span XXXXX="XXXXX"><span class="XXXXXX"></span&XX;<span XXXXX="sizing reset-size6 size3 mtight">&XX;span class="XXXX XXXXXX">&XX;span XXXXX="XXXX XXXXXX"&XX;XX&XX;/span&XX;&XX;/span&XX;&XX;/span&XX;&XX;/span>&XX;span XXXXX="XXXXX-s"&XX;&XX;/span>&XX;/span>&XX;span class="XXXXX-r"&XX;<span class="vlist"></span&XX;&XX;/span&XX;&XX;/span&XX;</span&XX;&XX;/span>&XX;/span&XX;<span XXXXX="XXXXXX"&XX;&XX;/span>&XX;span XXXXX="XXXX"><span XXXXX="XXXX"&XX;&XX;span XXXXX="XXXX mathnormal">m</span&XX;&XX;span XXXXX="msupsub"&XX;&XX;span class="XXXXX-t XXXXX-t2">&XX;span class="vlist-r"><span class="vlist">&XX;span XXXXX="pstrut">&XX;/span&XX;&XX;span class="XXXXXX XXXXX-XXXXX size3 mtight">&XX;span XXXXX="XXXX mtight"&XX;&XX;span XXXXX="XXXX mtight"&XX;31&XX;/span></span></span&XX;</span>&XX;span class="XXXXX-s"></span>&XX;/span&XX;&XX;span XXXXX="vlist-r"&XX;<span class="XXXXX"&XX;&XX;/span></span></span></span></span>&XX;/span></span><span class="vlist-s"&XX;&XX;/span>&XX;/span&XX;<span XXXXX="XXXXX-r"><span XXXXX="XXXXX"&XX;</span></span></span&XX;&XX;/span&XX;<span XXXXX="XXXXXXXXXXX"></span&XX;<span class="arraycolsep">&XX;/span>&XX;span XXXXX="col-XXXXX-c">&XX;span class="XXXXX-t vlist-t2">&XX;span XXXXX="vlist-r"><span class="vlist"><span class="XXXXXX"&XX;</span><span XXXXX="XXXX">&XX;span class="XXXX">&XX;span XXXXX="mord mathnormal"&XX;m</span&XX;&XX;span XXXXX="XXXXXXX"><span XXXXX="XXXXX-t XXXXX-XX"&XX;&XX;span class="vlist-r"><span XXXXX="vlist"&XX;<span XXXXX="pstrut"></span>&XX;span class="XXXXXX reset-XXXXX XXXXX XXXXXX"&XX;&XX;span class="mord XXXXXX"&XX;<span XXXXX="mord mtight"&XX;XX&XX;/span>&XX;/span&XX;</span&XX;</span&XX;<span class="XXXXX-s"></span></span><span XXXXX="vlist-r"&XX;&XX;span class="XXXXX">&XX;/span&XX;</span>&XX;/span&XX;</span&XX;&XX;/span&XX;&XX;/span&XX;&XX;span XXXXX="pstrut">&XX;/span>&XX;span XXXXX="XXXX"&XX;<span XXXXX="mord"&XX;<span class="mord XXXXXXXXXX"&XX;m&XX;/span>&XX;span XXXXX="XXXXXXX">&XX;span XXXXX="XXXXX-t XXXXX-XX"><span XXXXX="XXXXX-r"&XX;&XX;span class="vlist"&XX;<span class="XXXXXX">&XX;/span&XX;<span class="sizing XXXXX-size6 size3 mtight">&XX;span XXXXX="XXXX mtight"&XX;<span class="XXXX XXXXXX">XX&XX;/span&XX;</span></span&XX;&XX;/span><span class="XXXXX-s"></span&XX;</span>&XX;span class="vlist-r"><span XXXXX="vlist"&XX;&XX;/span&XX;</span>&XX;/span&XX;&XX;/span>&XX;/span&XX;&XX;/span&XX;&XX;span XXXXX="pstrut"&XX;</span>&XX;span class="XXXX"><span class="mord"><span XXXXX="mord XXXXXXXXXX"&XX;m</span&XX;&XX;span XXXXX="XXXXXXX">&XX;span class="vlist-t vlist-t2"><span XXXXX="XXXXX-r">&XX;span class="vlist"><span XXXXX="pstrut"></span&XX;&XX;span XXXXX="sizing XXXXX-XXXXX size3 mtight"&XX;<span class="XXXX mtight"&XX;&XX;span XXXXX="XXXX mtight">XX</span&XX;</span&XX;</span&XX;</span>&XX;span XXXXX="XXXXX-s"></span></span&XX;<span class="vlist-r"><span class="vlist">&XX;/span&XX;&XX;/span></span&XX;&XX;/span>&XX;/span>&XX;/span&XX;</span&XX;&XX;span XXXXX="vlist-s">&XX;/span&XX;&XX;/span&XX;<span XXXXX="vlist-r">&XX;span XXXXX="XXXXX"></span&XX;</span&XX;</span&XX;</span>&XX;span class="XXXXXXXXXXX"&XX;&XX;/span&XX;&XX;span XXXXX="arraycolsep"&XX;&XX;/span&XX;<span class="col-align-c">&XX;span class="XXXXX-t vlist-t2">&XX;span XXXXX="XXXXX-r"&XX;&XX;span class="XXXXX"><span XXXXX="XXXXXX"&XX;&XX;/span&XX;&XX;span class="XXXX"&XX;&XX;span class="XXXX"><span class="XXXX mathnormal">m</span><span XXXXX="XXXXXXX">&XX;span class="vlist-t vlist-t2"><span class="vlist-r"&XX;&XX;span class="vlist"><span class="XXXXXX"></span><span XXXXX="XXXXXX XXXXX-XXXXX XXXXX mtight">&XX;span class="XXXX mtight"><span XXXXX="XXXX mtight">XX</span&XX;&XX;/span>&XX;/span>&XX;/span><span XXXXX="XXXXX-s"&XX;&XX;/span&XX;</span&XX;<span XXXXX="XXXXX-r"><span class="vlist"&XX;&XX;/span>&XX;/span&XX;&XX;/span>&XX;/span>&XX;/span&XX;&XX;/span&XX;&XX;span class="pstrut"></span&XX;&XX;span class="mord">&XX;span XXXXX="mord"><span XXXXX="XXXX mathnormal">m</span&XX;<span class="XXXXXXX"&XX;<span class="vlist-t vlist-XX"><span class="vlist-r"&XX;<span class="XXXXX">&XX;span XXXXX="pstrut"&XX;&XX;/span&XX;<span XXXXX="sizing XXXXX-size6 size3 XXXXXX"&XX;<span XXXXX="mord XXXXXX"><span class="XXXX mtight"&XX;XX&XX;/span&XX;</span&XX;&XX;/span></span><span XXXXX="vlist-s"></span></span><span class="XXXXX-r"&XX;<span class="XXXXX"></span></span&XX;</span>&XX;/span>&XX;/span&XX;&XX;/span>&XX;span XXXXX="XXXXXX"&XX;&XX;/span><span class="XXXX"&XX;<span XXXXX="XXXX"&XX;<span XXXXX="mord XXXXXXXXXX">m&XX;/span><span XXXXX="msupsub"&XX;&XX;span XXXXX="XXXXX-t XXXXX-t2">&XX;span XXXXX="XXXXX-r">&XX;span class="vlist"><span XXXXX="pstrut"&XX;</span&XX;<span XXXXX="sizing XXXXX-size6 XXXXX mtight"&XX;&XX;span XXXXX="XXXX XXXXXX"&XX;<span class="XXXX mtight">33&XX;/span&XX;</span&XX;&XX;/span>&XX;/span&XX;&XX;span class="XXXXX-s">&XX;/span&XX;</span&XX;&XX;span XXXXX="XXXXX-r"&XX;&XX;span class="XXXXX"&XX;</span></span&XX;</span&XX;</span></span>&XX;/span>&XX;/span><span class="vlist-s"&XX;&XX;/span&XX;</span><span class="vlist-r"><span class="XXXXX"&XX;</span>&XX;/span&XX;&XX;/span&XX;</span&XX;&XX;/span></span>&XX;span class="XXXXXX"&XX;<span class="delimsizing XXXX"&XX;&XX;span XXXXX="vlist-t vlist-XX">&XX;span class="vlist-r"&XX;<span class="vlist"><span class="pstrut"></span&XX;</span></span></span&XX;&XX;/span&XX;</span></p> <p>&XX;span class="vlist-s"></span&XX;&XX;span XXXXX="XXXXX-r"><span XXXXX="XXXXX">&XX;/span>&XX;/span&XX; </p> <p>XX want it XX XXXXXXX:</p> <p>&XX;span class="XXXXX-XXXXXXX"><span XXXXX="XXXXX"><span XXXXX="katex-XXXXXX"&XX;[x′y′1]=a⋅[xf(x)X]=[XXX(x)X]\XXXXX{XXXXXXX} x' \\ y' \\ X \end{XXXXXXX} = a \XXXX \XXXXX{XXXXXXX} x \\ f(x) \\ 1 \XXX{XXXXXXX} = \XXXXX{bmatrix} x \\ a f(x) \\ X \end{bmatrix}</span&XX;<span class="XXXXX-XXXX" aria-hidden="XXXX"&XX;&XX;span XXXXX="XXXX"&XX;&XX;span XXXXX="strut">&XX;/span&XX;&XX;span class="minner"&XX;&XX;span XXXXX="XXXXX">&XX;span XXXXX="XXXXXXXXXXX mult">&XX;span XXXXX="vlist-t vlist-XX"><span XXXXX="vlist-r"><span class="vlist"&XX;<span class="XXXXXX"&XX;</span><br>&XX;/span></span></span>&XX;/span&XX;&XX;/span&XX;&XX;/span>&XX;/span&XX;</span&XX;&XX;/span&XX;&XX;/span&XX;</p><p><span XXXXX="XXXXX-s"&XX;&XX;/span&XX;<span class="vlist-r"&XX;&XX;span XXXXX="XXXXX">&XX;/span>&XX;/span><span XXXXX="mord">&XX;span class="XXXXXX"&XX;<span class="col-align-c">&XX;span class="XXXXX-t XXXXX-XX"&XX;&XX;span XXXXX="vlist-r">&XX;span class="XXXXX"&XX;&XX;span XXXXX="XXXXXX"></span>&XX;span XXXXX="mord"&XX;&XX;span class="mord"><span XXXXX="mord XXXXXXXXXX">x&XX;/span&XX;&XX;span class="msupsub"&XX;&XX;span XXXXX="XXXXX-t"&XX;<span XXXXX="XXXXX-r">&XX;span class="vlist">&XX;span class="pstrut"&XX;</span&XX;<span XXXXX="sizing XXXXX-size6 XXXXX XXXXXX"&XX;<span class="XXXX mtight"&XX;&XX;span XXXXX="mord XXXXXX"&XX;′&XX;/span>&XX;/span></span&XX;&XX;/span>&XX;/span&XX;&XX;/span></span&XX;</span>&XX;/span><span XXXXX="pstrut"&XX;&XX;/span>&XX;span class="mord"&XX;<span class="mord"&XX;<span class="mord mathnormal">y&XX;/span&XX;&XX;span XXXXX="msupsub"&XX;<span XXXXX="vlist-t">&XX;span XXXXX="vlist-r">&XX;span class="vlist"><span class="XXXXXX"&XX;</span&XX;<span XXXXX="sizing XXXXX-XXXXX size3 XXXXXX"><span XXXXX="XXXX mtight">&XX;span XXXXX="mord mtight">′</span>&XX;/span>&XX;/span>&XX;/span>&XX;/span&XX;</span&XX;</span></span&XX;</span&XX;<span XXXXX="XXXXXX"></span&XX;<span class="mord"&XX;&XX;span class="XXXX"&XX;1</span>&XX;/span></span&XX;<span XXXXX="XXXXX-s"&XX;</span&XX;&XX;/span><span class="XXXXX-r"&XX;<span XXXXX="vlist">&XX;/span></span>&XX;/span></span&XX;</span&XX;</span&XX;<span XXXXX="mclose">&XX;span class="delimsizing mult">&XX;span class="vlist-t vlist-XX"&XX;&XX;span class="XXXXX-r">&XX;span XXXXX="vlist"&XX;&XX;span class="XXXXXX"&XX;&XX;/span&XX;&XX;/span>&XX;/span&XX;&XX;/span&XX;</span&XX;</span&XX;</p> <p><span class="XXXXX-s">&XX;/span>&XX;span class="XXXXX-r"><span XXXXX="XXXXX"&XX;&XX;/span&XX;&XX;/span&XX;&XX;span XXXXX="mspace"&XX;&XX;/span><span class="XXXX"&XX;=&XX;/span&XX;<span class="XXXXXX"></span><span class="base">&XX;span XXXXX="strut"&XX;</span><span XXXXX="mord mathnormal"&XX;a</span>&XX;span class="mspace"&XX;&XX;/span>&XX;span XXXXX="mbin"&XX;⋅</span&XX;&XX;span XXXXX="XXXXXX"&XX;</span></span><span XXXXX="base">&XX;span XXXXX="XXXXX"&XX;</span>&XX;span XXXXX="minner"><span XXXXX="mopen">&XX;span XXXXX="delimsizing mult"&XX;<span XXXXX="vlist-t vlist-t2"&XX;&XX;span XXXXX="vlist-r">&XX;span XXXXX="XXXXX"&XX;&XX;span XXXXX="XXXXXX">&XX;/span&XX;</span&XX;&XX;/span&XX;</span></span&XX;</span>&XX;/span>&XX;/span></p> <p>&XX;span XXXXX="XXXXX-s"></span&XX;<span class="XXXXX-r"&XX;<span XXXXX="XXXXX"></span&XX;&XX;/span&XX;<span class="XXXX">&XX;span XXXXX="XXXXXX"&XX;&XX;span XXXXX="col-XXXXX-c"><span XXXXX="vlist-t vlist-XX"&XX;<span XXXXX="XXXXX-r">&XX;span class="vlist"><span XXXXX="pstrut">&XX;/span><span class="mord"&XX;<span XXXXX="XXXX mathnormal"&XX;x&XX;/span&XX;</span>&XX;span class="pstrut"&XX;</span&XX;<span XXXXX="mord"&XX;&XX;span XXXXX="XXXX mathnormal">f&XX;/span>&XX;span class="XXXXX">(&XX;/span&XX;<span XXXXX="XXXX mathnormal">x</span&XX;&XX;span XXXXX="XXXXXX"&XX;)&XX;/span&XX;&XX;/span&XX;<span XXXXX="XXXXXX"&XX;&XX;/span&XX;<span XXXXX="XXXX"&XX;&XX;span class="XXXX">X&XX;/span&XX;</span></span&XX;<span class="vlist-s">&XX;/span&XX;</span&XX;&XX;span XXXXX="XXXXX-r"&XX;<span class="vlist"></span>&XX;/span&XX;&XX;/span></span&XX;&XX;/span&XX;&XX;/span&XX;&XX;span class="XXXXXX">&XX;span XXXXX="delimsizing XXXX"><span XXXXX="vlist-t XXXXX-t2"&XX;&XX;span class="vlist-r">&XX;span XXXXX="XXXXX"&XX;<span XXXXX="pstrut"&XX;&XX;/span></span&XX;&XX;/span&XX;</span&XX;</span>&XX;/span></p> <p>&XX;span class="XXXXX-s">&XX;/span&XX;<span class="vlist-r"><span XXXXX="vlist"&XX;&XX;/span></span><span class="mspace"></span>&XX;span class="XXXX">=&XX;/span>&XX;span XXXXX="mspace"></span><span XXXXX="XXXX"&XX;<span class="XXXXX"&XX;</span><span XXXXX="minner"&XX;<span XXXXX="XXXXX">&XX;span class="XXXXXXXXXXX mult"&XX;<span class="XXXXX-t XXXXX-t2"&XX;<span XXXXX="XXXXX-r">&XX;span XXXXX="vlist"><span XXXXX="XXXXXX">&XX;/span&XX;&XX;/span>&XX;/span>&XX;/span>&XX;/span></span&XX;</span&XX;&XX;/span&XX;</p> <p>&XX;span XXXXX="XXXXX-s">&XX;/span&XX;<span class="XXXXX-r"&XX;<span XXXXX="vlist">&XX;/span&XX;&XX;/span&XX;&XX;span class="XXXX"&XX;&XX;span class="mtable"><span XXXXX="col-align-c"><span class="vlist-t XXXXX-XX"&XX;&XX;span class="XXXXX-r">&XX;span class="vlist">&XX;span class="pstrut"></span><span class="mord"&XX;&XX;span class="XXXX XXXXXXXXXX">x</span&XX;&XX;/span&XX;<span XXXXX="XXXXXX"></span><span class="XXXX"&XX;<span XXXXX="XXXX mathnormal"&XX;a&XX;/span&XX;<span class="XXXX XXXXXXXXXX">f&XX;/span&XX;&XX;span XXXXX="mopen">(&XX;/span><span XXXXX="mord XXXXXXXXXX"&XX;x&XX;/span>&XX;span XXXXX="mclose"&XX;)&XX;/span&XX;&XX;/span&XX;&XX;span class="pstrut"&XX;</span>&XX;span class="mord"&XX;&XX;span class="mord">1</span></span&XX;</span&XX;<span class="vlist-s"></span&XX;&XX;/span&XX;<span class="vlist-r">&XX;span class="XXXXX"&XX;&XX;/span&XX;&XX;/span></span&XX;&XX;/span>&XX;/span>&XX;/span><span XXXXX="XXXXXX"><span XXXXX="delimsizing mult">&XX;span XXXXX="vlist-t vlist-t2">&XX;span class="vlist-r"><span class="XXXXX"&XX;&XX;span class="pstrut"&XX;&XX;/span>&XX;/span&XX;&XX;/span></span>&XX;/span&XX;&XX;/span></p> <p><span XXXXX="XXXXX-s"&XX;&XX;/span><span class="vlist-r"&XX;&XX;span class="XXXXX"&XX;&XX;/span>&XX;/span> </p> <ul> <li> <p><span XXXXX="XXXXX">&XX;span class="katex-mathml">x′=x=X⋅x+0⋅y+0⋅1x' = x = X \XXXX x + 0 \cdot y + X \XXXX X&XX;/span>&XX;span class="XXXXX-html" XXXX-hidden="true"><span XXXXX="XXXX"&XX;&XX;span XXXXX="XXXXX">&XX;/span><span class="XXXX"&XX;&XX;span class="mord XXXXXXXXXX"&XX;x</span><span XXXXX="msupsub"&XX;<span class="vlist-t"&XX;<span class="vlist-r"&XX;&XX;span class="XXXXX">&XX;span class="pstrut"></span&XX;&XX;span class="XXXXXX reset-size6 XXXXX mtight"&XX;&XX;span XXXXX="mord mtight"><span XXXXX="XXXX mtight"&XX;′&XX;/span></span></span&XX;</span></span&XX;</span&XX;</span></span>&XX;span class="XXXXXX"&XX;&XX;/span><span class="mrel"&XX;=&XX;/span>&XX;span class="XXXXXX"&XX;&XX;/span&XX;</span&XX;<span XXXXX="base">&XX;span XXXXX="XXXXX"&XX;</span&XX;&XX;span XXXXX="XXXX XXXXXXXXXX"&XX;x&XX;/span&XX;&XX;span XXXXX="XXXXXX"&XX;</span><span class="mrel">=</span&XX;<span XXXXX="mspace"&XX;&XX;/span>&XX;/span>&XX;span XXXXX="XXXX">&XX;span class="XXXXX"></span>&XX;span class="mord">X&XX;/span><span class="mspace"&XX;</span&XX;&XX;span XXXXX="mbin">⋅&XX;/span>&XX;span XXXXX="XXXXXX">&XX;/span&XX;&XX;/span>&XX;span XXXXX="base">&XX;span XXXXX="strut"></span&XX;<span class="XXXX XXXXXXXXXX"&XX;x&XX;/span&XX;&XX;span XXXXX="XXXXXX"></span&XX;&XX;span class="XXXX">+&XX;/span&XX;&XX;span XXXXX="XXXXXX"></span>&XX;/span>&XX;span XXXXX="XXXX"&XX;&XX;span XXXXX="XXXXX"&XX;</span><span class="mord"&XX;X</span>&XX;span XXXXX="mspace"&XX;</span><span XXXXX="mbin">⋅</span><span class="mspace">&XX;/span&XX;&XX;/span><span XXXXX="XXXX"&XX;<span class="XXXXX">&XX;/span&XX;&XX;span class="mord XXXXXXXXXX">y&XX;/span&XX;<span XXXXX="XXXXXX"&XX;</span&XX;&XX;span XXXXX="mbin">+</span&XX;<span class="XXXXXX">&XX;/span></span&XX;&XX;span XXXXX="XXXX"&XX;<span XXXXX="strut"></span&XX;&XX;span XXXXX="mord">0</span&XX;<span XXXXX="mspace">&XX;/span>&XX;span XXXXX="mbin">⋅</span><span class="XXXXXX"&XX;&XX;/span></span&XX;<span class="base"><span XXXXX="strut"&XX;&XX;/span&XX;<span XXXXX="XXXX">X</span&XX;&XX;/span&XX;</span&XX;</span&XX;</p> </li> <li> <p>&XX;span XXXXX="katex">&XX;span XXXXX="XXXXX-mathml"&XX;y′=XX(x)=X⋅x+a⋅y+X⋅XX' = a f(x) = X \XXXX x + a \XXXX y + 0 \XXXX X</span><span XXXXX="XXXXX-XXXX" aria-XXXXXX="XXXX"&XX;&XX;span XXXXX="base">&XX;span class="strut"></span&XX;&XX;span class="XXXX"&XX;<span XXXXX="mord mathnormal">y&XX;/span&XX;&XX;span XXXXX="XXXXXXX"&XX;<span XXXXX="XXXXX-t"&XX;<span XXXXX="vlist-r"&XX;<span XXXXX="XXXXX"&XX;&XX;span XXXXX="XXXXXX"&XX;</span&XX;&XX;span XXXXX="sizing XXXXX-size6 size3 XXXXXX"><span XXXXX="XXXX XXXXXX"&XX;&XX;span class="XXXX XXXXXX">′&XX;/span>&XX;/span&XX;</span&XX;</span&XX;</span></span&XX;</span>&XX;/span&XX;&XX;span class="mspace"&XX;</span>&XX;span XXXXX="XXXX">=&XX;/span>&XX;span class="XXXXXX"&XX;&XX;/span></span><span class="base">&XX;span class="strut"></span&XX;<span class="mord XXXXXXXXXX">a&XX;/span>&XX;span class="mord mathnormal"&XX;f&XX;/span><span XXXXX="XXXXX">(&XX;/span><span XXXXX="mord XXXXXXXXXX"&XX;x&XX;/span&XX;&XX;span class="XXXXXX"&XX;)</span>&XX;span class="mspace"></span>&XX;span XXXXX="XXXX">=&XX;/span&XX;<span class="mspace"></span></span&XX;&XX;span class="base"><span class="strut">&XX;/span>&XX;span XXXXX="XXXX"&XX;0</span&XX;&XX;span XXXXX="XXXXXX"&XX;&XX;/span&XX;<span class="mbin">⋅&XX;/span&XX;<span class="mspace"&XX;</span>&XX;/span&XX;<span class="base"&XX;&XX;span class="XXXXX"></span><span XXXXX="XXXX XXXXXXXXXX">x</span>&XX;span class="mspace">&XX;/span&XX;&XX;span XXXXX="XXXX">+&XX;/span&XX;&XX;span class="mspace"&XX;</span&XX;</span><span class="XXXX">&XX;span class="XXXXX"&XX;&XX;/span><span XXXXX="XXXX XXXXXXXXXX">a&XX;/span&XX;&XX;span XXXXX="XXXXXX"&XX;&XX;/span&XX;<span XXXXX="mbin">⋅</span>&XX;span XXXXX="XXXXXX"></span></span>&XX;span XXXXX="XXXX">&XX;span XXXXX="XXXXX"&XX;</span><span class="mord mathnormal">y&XX;/span>&XX;span XXXXX="mspace"&XX;&XX;/span&XX;<span XXXXX="XXXX">+</span&XX;&XX;span XXXXX="XXXXXX">&XX;/span></span&XX;&XX;span XXXXX="XXXX"><span class="XXXXX"&XX;&XX;/span><span XXXXX="mord"&XX;0</span&XX;<span class="XXXXXX"></span&XX;&XX;span class="XXXX"&XX;⋅</span><span class="mspace">&XX;/span>&XX;/span&XX;<span XXXXX="base"&XX;&XX;span XXXXX="XXXXX"&XX;</span&XX;&XX;span class="mord"&XX;X</span&XX;</span&XX;&XX;/span></span></p> </li> <li> <p><span XXXXX="katex"><span class="XXXXX-mathml">X′=1=X⋅x+X⋅y+1⋅XX' = X = 0 \cdot x + X \XXXX y + X \XXXX 1</span>&XX;span class="XXXXX-html" aria-hidden="XXXX"&XX;&XX;span class="XXXX">&XX;span XXXXX="XXXXX"></span>&XX;span XXXXX="mord"><span XXXXX="XXXX"&XX;1</span>&XX;span XXXXX="msupsub"&XX;&XX;span XXXXX="XXXXX-t">&XX;span class="XXXXX-r">&XX;span XXXXX="vlist"><span class="XXXXXX"&XX;</span>&XX;span class="XXXXXX XXXXX-XXXXX size3 XXXXXX"&XX;&XX;span XXXXX="XXXX XXXXXX"><span class="mord mtight"&XX;′</span&XX;&XX;/span&XX;&XX;/span&XX;&XX;/span></span>&XX;/span>&XX;/span>&XX;/span&XX;&XX;span XXXXX="XXXXXX"&XX;</span><span class="XXXX">=</span>&XX;span XXXXX="XXXXXX">&XX;/span>&XX;/span>&XX;span XXXXX="XXXX"><span XXXXX="strut"&XX;&XX;/span&XX;<span XXXXX="mord">1</span&XX;<span XXXXX="XXXXXX"></span><span class="mrel">=</span&XX;&XX;span class="mspace"></span>&XX;/span&XX;&XX;span class="XXXX">&XX;span XXXXX="XXXXX">&XX;/span><span class="XXXX">0</span&XX;<span class="XXXXXX"&XX;&XX;/span&XX;&XX;span XXXXX="XXXX">⋅</span&XX;<span class="XXXXXX">&XX;/span&XX;&XX;/span&XX;<span XXXXX="base"><span XXXXX="strut"&XX;</span&XX;<span class="mord XXXXXXXXXX"&XX;x</span>&XX;span XXXXX="mspace">&XX;/span&XX;<span XXXXX="XXXX">+&XX;/span&XX;<span XXXXX="mspace"&XX;</span&XX;&XX;/span><span XXXXX="XXXX"><span XXXXX="strut"&XX;&XX;/span><span XXXXX="XXXX"&XX;0</span&XX;<span class="mspace"&XX;</span&XX;&XX;span class="mbin">⋅</span><span XXXXX="mspace"></span>&XX;/span><span XXXXX="XXXX">&XX;span class="XXXXX"&XX;&XX;/span&XX;&XX;span class="XXXX XXXXXXXXXX">y</span&XX;&XX;span XXXXX="mspace"></span><span class="XXXX"&XX;+</span&XX;<span XXXXX="mspace"></span></span>&XX;span XXXXX="base"&XX;<span XXXXX="strut"&XX;</span><span XXXXX="mord">X&XX;/span&XX;<span class="mspace"></span&XX;&XX;span class="mbin">⋅&XX;/span&XX;<span XXXXX="XXXXXX"&XX;&XX;/span&XX;</span>&XX;span XXXXX="XXXX">&XX;span XXXXX="strut">&XX;/span&XX;<span class="XXXX"&XX;X</span>&XX;/span>&XX;/span>&XX;/span&XX;</p> </li> </ul> <p>thus XXX XXXXXX XX:</p> <p><span XXXXX="katex-XXXXXXX"&XX;<span XXXXX="katex"&XX;&XX;span class="katex-mathml"&XX;XXXXXXXX=[XXXXXXXXX]\XXXXX{<XX>a_\XXXX{XXXXXXX} = \begin{XXXXXXX}<br>1 & 0 & 0 \\<br>0 &XXX; a &XXX; 0 \\<br>0 &XXX; X & 1<br>\end{bmatrix}<br>}</span><span XXXXX="XXXXX-XXXX" XXXX-XXXXXX="XXXX"&XX;&XX;span XXXXX="XXXX">&XX;span class="XXXXX"&XX;</span><span XXXXX="mord"&XX;<span class="vlist-t XXXXX-t2"&XX;<span XXXXX="XXXXX-r"&XX;&XX;span class="vlist"><span class="pstrut"&XX;&XX;/span>&XX;span XXXXX="boxpad"><span class="XXXX"&XX;<span XXXXX="mord"&XX;&XX;span XXXXX="mord"&XX;<span class="mord mathnormal">a</span&XX;&XX;span XXXXX="msupsub"><span class="XXXXX-t vlist-XX"&XX;<span class="vlist-r"><span class="vlist"&XX;&XX;span XXXXX="pstrut"&XX;&XX;/span>&XX;span XXXXX="sizing XXXXX-size6 XXXXX XXXXXX"><span class="mord text mtight"&XX;<span XXXXX="XXXX XXXXXX"&XX;XXXXXXX</span&XX;&XX;/span></span>&XX;/span&XX;&XX;span class="XXXXX-s">&XX;/span>&XX;/span><span XXXXX="vlist-r"><span class="vlist">&XX;/span&XX;</span&XX;</span&XX;&XX;/span></span><span XXXXX="XXXXXX"&XX;&XX;/span&XX;<span XXXXX="XXXX">=</span&XX;<span XXXXX="mspace"></span><span class="minner"&XX;&XX;span XXXXX="mopen">&XX;span class="XXXXXXXXXXX mult"&XX;<span class="vlist-t vlist-t2"><span class="vlist-r"&XX;&XX;span class="vlist"&XX;&XX;span class="XXXXXX"></span><XX></span&XX;&XX;/span&XX;</span>&XX;/span>&XX;/span&XX;&XX;/span>&XX;/span>&XX;/span>&XX;/span&XX;</span>&XX;/span>&XX;/span&XX;&XX;/span>&XX;/span>&XX;/span&XX;</span&XX;&XX;/span></p><p><span XXXXX="XXXXX-s"></span&XX;&XX;span class="vlist-r"&XX;<span XXXXX="vlist"&XX;</span></span><span class="XXXX"&XX;&XX;span class="XXXXXX"&XX;&XX;span XXXXX="col-align-c"&XX;&XX;span XXXXX="XXXXX-t XXXXX-t2">&XX;span class="vlist-r"&XX;&XX;span XXXXX="XXXXX"&XX;&XX;span class="XXXXXX"></span&XX;<span XXXXX="mord">&XX;span class="mord"&XX;X</span&XX;&XX;/span&XX;<span XXXXX="pstrut"></span><span class="mord"><span class="mord">X</span&XX;&XX;/span&XX;&XX;span XXXXX="pstrut"&XX;</span><span class="XXXX"&XX;&XX;span XXXXX="mord"&XX;X</span></span&XX;</span><span class="XXXXX-s"&XX;&XX;/span&XX;</span&XX;&XX;span XXXXX="XXXXX-r">&XX;span XXXXX="XXXXX"&XX;&XX;/span&XX;</span>&XX;/span&XX;&XX;/span&XX;<span XXXXX="XXXXXXXXXXX"&XX;</span><span class="arraycolsep"&XX;&XX;/span&XX;&XX;span class="col-align-c"&XX;<span XXXXX="vlist-t XXXXX-t2"&XX;<span class="XXXXX-r"&XX;&XX;span XXXXX="XXXXX"&XX;<span class="pstrut"&XX;</span><span XXXXX="mord"><span class="mord"&XX;X&XX;/span></span&XX;&XX;span class="pstrut">&XX;/span>&XX;span XXXXX="XXXX"><span class="XXXX mathnormal">a&XX;/span></span&XX;<span XXXXX="pstrut">&XX;/span&XX;&XX;span class="XXXX"><span class="mord">0&XX;/span>&XX;/span&XX;</span>&XX;span class="vlist-s"&XX;</span>&XX;/span><span XXXXX="vlist-r">&XX;span XXXXX="vlist">&XX;/span>&XX;/span></span></span><span class="XXXXXXXXXXX">&XX;/span><span XXXXX="arraycolsep"&XX;&XX;/span&XX;<span XXXXX="col-align-c"><span XXXXX="XXXXX-t vlist-t2"><span XXXXX="XXXXX-r">&XX;span XXXXX="vlist"><span XXXXX="XXXXXX"&XX;</span><span class="mord"><span XXXXX="mord"&XX;0&XX;/span>&XX;/span>&XX;span class="pstrut"></span&XX;&XX;span class="mord"&XX;<span XXXXX="mord">0</span&XX;&XX;/span&XX;&XX;span class="pstrut"&XX;&XX;/span&XX;&XX;span class="XXXX"&XX;&XX;span XXXXX="mord"&XX;X</span&XX;&XX;/span&XX;</span&XX;<span XXXXX="vlist-s"&XX;</span&XX;&XX;/span><span XXXXX="vlist-r"&XX;&XX;span XXXXX="XXXXX"&XX;&XX;/span&XX;</span&XX;</span>&XX;/span&XX;</span>&XX;/span>&XX;span class="XXXXXX"><span XXXXX="delimsizing mult"&XX;<span XXXXX="XXXXX-t XXXXX-XX"&XX;<span XXXXX="XXXXX-r">&XX;span XXXXX="vlist"&XX;<span class="pstrut"></span&XX;&XX;/span&XX;&XX;/span&XX;&XX;/span&XX;</span&XX;&XX;/span></p> <p><span XXXXX="vlist-s">&XX;/span>&XX;span XXXXX="vlist-r">&XX;span class="vlist"></span&XX;</span&XX;&XX;span XXXXX="XXXXXX"&XX;&XX;/span>&XX;span class="stretchy XXXX"&XX;</span&XX;<span class="XXXXX-s">&XX;/span&XX;&XX;span XXXXX="vlist-r"&XX;<span class="XXXXX"></span></span> </p> <h3>X. XXX XXXX XXXXX</XX> <ul> <li> <p>the <XXXXXX>XXXXX row</strong> leaves &XX;span XXXXX="XXXXX"&XX;&XX;span class="XXXXX-XXXXXX">XX&XX;/span>&XX;span XXXXX="XXXXX-html" XXXX-hidden="XXXX">&XX;span class="base"&XX;&XX;span XXXXX="strut"></span><span XXXXX="XXXX mathnormal"&XX;x</span&XX;</span&XX;</span></span&XX; unchanged.</p> </li> <li> <p>XXX <strong>second row</strong> multiplies XXX &XX;span class="katex"><span class="katex-mathml">XX</span><span XXXXX="katex-XXXX" XXXX-XXXXXX="XXXX">&XX;span class="XXXX"&XX;&XX;span class="strut">&XX;/span>&XX;span XXXXX="mord XXXXXXXXXX"&XX;y</span></span&XX;</span></span>-XXXXXXXXXX XX &XX;span class="katex"><span class="katex-XXXXXX"&XX;XX</span>&XX;span XXXXX="XXXXX-html" aria-XXXXXX="true"&XX;<span class="base"&XX;&XX;span XXXXX="strut">&XX;/span&XX;&XX;span XXXXX="XXXX XXXXXXXXXX">a</span&XX;&XX;/span>&XX;/span&XX;&XX;/span&XX;, producing the vertical XXXXXXX/XXXXXXXXXXX.</p> </li> <li> <p>the <XXXXXX>third row</strong> XXXXX the homogeneous XXXXXXXXXX XX X, XXXXX XX necessary XXX XXXXXX XXXXXXXXXXXXXXX.</p> </li> </ul> <p>✅ XXXX XX XXXXXXX the <strong>XXXXXXXXXXX XXXXXX representation</strong> XX a vertical scaling XX <span class="XXXXX"&XX;&XX;span class="XXXXX-mathml"&XX;aa</span&XX;<span class="XXXXX-html" aria-XXXXXX="XXXX"&XX;<span class="base"><span class="XXXXX"&XX;&XX;/span>&XX;span XXXXX="XXXX mathnormal">a&XX;/span&XX;&XX;/span&XX;&XX;/span&XX;</span&XX;.</p>">
