Commit bb5636b0 by Jim Hefferon

### three_iv

parent be4278d5
 ... ... @@ -55,27 +55,24 @@ \section{Sums and Scalar Products} %.......... \begin{frame}{Representing operations on linear functions} Recall that the collection of linear functions between two spaces $\linmaps{V}{W}$ is a vector space. That is, given Recall that given a linear map $\map{f}{V}{W}$ then the scalar multiple $rf$ the scalar multiple function \begin{equation*} \vec{v}\mapsunder{r\cdot f} r\cdot (\,f(\vec{v})\,) \end{equation*} is a linear map $\map{rf}{V}{W}$. And, where $\map{f,g}{V}{W}$ are linear then their sum And where $\map{f,g}{V}{W}$ are linear then the function that is their sum \begin{equation*} \vec{v}\mapsunder{f+g} f(\vec{v})+g(\vec{v}) \end{equation*} is also linear $\map{f+g}{V}{W}$. \pause We now see how the matrix representation of $\rep{f}{B,D}$ is related to that of $\rep{rf}{B,D}$, and how the representations of and also how the representations of $\rep{f}{B,D}$ and $\rep{g}{B,D}$ combine to give the representation of $\rep{f+g}{B,D}$. \end{frame} ... ... @@ -83,15 +80,16 @@ combine to give the representation of $\rep{f+g}{B,D}$. \ex Fix a domain~$V$ and codomain~$W$ with bases $B=\sequence{\vec{\beta}_1,\vec{\beta}_2}$ and $D=\sequence{\vec{\delta}_1,\vec{\delta}_2}$. Let $\map{f}{V}{W}$ be the linear map represented by a matrix~$H$ and $D=\sequence{\vec{\delta}_1,\vec{\delta}_2}$, and let $\map{f}{V}{W}$ be the linear map represented by a matrix~$H$. We will find the matrix representing the map $\map{6f}{V}{W}$. \begin{equation*} \vec{v}\mapsunder{6f} 6\cdot f(\vec{v}) \end{equation*} Let this be the representation \pause Suppose that this is the representation of an output vector~$f(\vec{v})$. \begin{equation*} % \rep{\vec{v}}{B}=\colvec{v_1 \\ v_2} ... ... @@ -102,7 +100,7 @@ of an output vector~$f(\vec{v})$. Note that $6\cdot f(\vec{v})=6\cdot (w_1\vec{\delta}_1+w_2\vec{\delta}_2) =(6w_1)\cdot\vec{\delta}_1+(6w_2)\cdot\vec{\delta}_2$, so $6f\:(\vec{v})$ has so the output vector $6f\:(\vec{v})$ has this representation. \begin{equation*} % \rep{6f(\vec{v})}{D} ... ... @@ -140,7 +138,7 @@ and this is the matrix-vector representation of the action of~$6f$. \colvec{v_1 \\ v_2} =\colvec{12v_1+6v_2 \\ 18v_1+24v_2} \end{equation*} So this is the matrix representing~$6f$. So this must be the matrix representing the function~$6f$. \begin{equation*} \rep{6f}{B,D} = ... ... @@ -227,6 +225,10 @@ so this is its matrix representation. \begin{equation*} \rep{f+g}{B,D}= \begin{mat} 2+5 &1+8 \\ 3+7 &4+6 \end{mat} =\begin{mat} 7 &9 \\ 10 &10 \end{mat} ... ... @@ -274,7 +276,8 @@ Then \end{mat} \end{equation*} None of these is defined: $A+B$, $B+A$, $B+C$, $C+B$, because the sizes don't match. $B+C$, $C+B$, because the sizes don't match, making it impossible to do an entry-by-entry sum. \end{frame} ... ... @@ -341,11 +344,11 @@ is composition. \ExecuteMetaData[../map4.tex]{pf:CompositionOfLinearMapsIsLinear} \qed \pause \medskip We next do an exploratory calculation to see how the matrix representations of the two functions combine to make the matrix representation of their composition. % \pause % \medskip % We next do an exploratory calculation to see how the matrix % representations of the two functions combine to make % the matrix representation of their composition. \end{frame} ... ... @@ -374,13 +377,14 @@ represented as here. % $V$ has dimension~$2$, % $W$ has dimension~$3$, % and $X$ has dimension~$2$. We want to see how these two representations combine to give the representation of the map We will do an explatory computation, to see how these two representations combine to give the representation of the composition $\map{\composed{g}{h}}{V}{X}$. \pause We start with the action of~$h$ on a~$\vec{v}\in V$. of~$h$ on~$\vec{v}\in V$. \begin{align*} \rep{h(\vec{v})}{C} &=\rep{h}{B,C}\cdot\rep{\vec{v}}{B} \\ ... ... @@ -394,11 +398,9 @@ of~$h$ on a~$\vec{v}\in V$. = \colvec{3v_1+v_2 \\ 2v_1+5v_2 \\ 4v_1+6v_2} \end{align*} Of course we represent application of~$h$ by doing the matrix vector multiplication. \end{frame} \begin{frame} \noindent Next apply $g$. \noindent Next, to that apply $g$. \begin{multline*} \rep{g}{C,D}\cdot \rep{h(\vec{v})}{C} = ... ... @@ -470,8 +472,7 @@ So here is how the two starting matrices combine. \end{frame} \begin{frame} \ex This product is not defined because the number of columns on the left must equal the number of rows on the right. This product \begin{equation*} \begin{mat} 1 &3 &-1 \\ ... ... @@ -483,6 +484,8 @@ the number of columns on the left must equal the number of rows on the right. 2 &2 &0 \end{mat} \end{equation*} is not defined because the number of columns on the left must equal the number of rows on the right. \pause \ex ... ... @@ -726,9 +729,9 @@ These two functions can be composed \end{equation*} because the codomain of~$f$ is the domain of~$g$. Observe about the order of the notation:~in An important observation about the order in which we write these things:~in writing the composition~$\composed{g}{h}$, the function $g$ is written first (meaning leftmost) the function $g$ is written first, that is, leftmost, but it is applied second. \begin{equation*} \vec{v}\mapsunder{h} h(\vec{v}) \mapsunder{g} g( h(\vec{v})) ... ...
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