Lösning till övning 3

SamverkanLinalgLIU

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Nuvarande version (15 augusti 2008 kl. 08.45) (redigera) (ogör)
 
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Rad 1: Rad 1:
Låt <math>\boldsymbol{u}=\underline{\boldsymbol{e}}X_1=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}</math> och <math>\boldsymbol{v}=\underline{\boldsymbol{e}}X_2=\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}</math>.
Låt <math>\boldsymbol{u}=\underline{\boldsymbol{e}}X_1=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}</math> och <math>\boldsymbol{v}=\underline{\boldsymbol{e}}X_2=\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}</math>.
Vi behöver summan
Vi behöver summan
-
<center><math>\boldsymbol{u}+\boldsymbol{v}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}+\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1+a_2}\\{b_1+b_2}\\{c_1+c_2}\end{pmatrix}</center></math>
+
<center><math>\boldsymbol{u}+\boldsymbol{v}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}+\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1+a_2}\\{b_1+b_2}\\{c_1+c_2}\end{pmatrix}</math></center>
och
och
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<math>\lambda\boldsymbol{u}=\lambda\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}.</math>
+
<center><math>
 +
\lambda\boldsymbol{u}=\lambda\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda
 +
a_1}\\{\lambda b_1}\\{\lambda
 +
c_1}\end{pmatrix}.
 +
</math></center>
 +
Avbildningen <math>G</math> är inte linjär, ty
 +
<center><math>1.\quad G(\boldsymbol{u}+\boldsymbol{v})\neq G(\boldsymbol{u})+G(\boldsymbol{v})\qquad\qquad 2.\quad G(\lambda\boldsymbol{u})\neq\lambda G(\boldsymbol{u}).</math></center>
 +
T.ex., följer att
 +
 
 +
<center><math>
 +
\begin{align}G(\lambda\boldsymbol{u})&=G\left(\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}\right)=G\left(\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}\right)=\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1\cdot\lambda c_1}\\{\lambda^2b_1^2}\\{\lambda b_1+\lambda c_1}\end{pmatrix}\\
 +
&=\lambda\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1c_1}\\{\lambda b_1^2}\\{b_1+c_1}\end{pmatrix}\neq\lambda G(\boldsymbol{u}).\end{align}</math></center>

Nuvarande version

Låt \displaystyle \boldsymbol{u}=\underline{\boldsymbol{e}}X_1=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix} och \displaystyle \boldsymbol{v}=\underline{\boldsymbol{e}}X_2=\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}. Vi behöver summan

\displaystyle \boldsymbol{u}+\boldsymbol{v}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}+\underline{\boldsymbol{e}}\begin{pmatrix}{a_2}\\{b_2}\\{c_2}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{a_1+a_2}\\{b_1+b_2}\\{c_1+c_2}\end{pmatrix}

och

\displaystyle

\lambda\boldsymbol{u}=\lambda\underline{\boldsymbol{e}}\begin{pmatrix}{a_1}\\{b_1}\\{c_1}\end{pmatrix}=\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}.

Avbildningen \displaystyle G är inte linjär, ty

\displaystyle 1.\quad G(\boldsymbol{u}+\boldsymbol{v})\neq G(\boldsymbol{u})+G(\boldsymbol{v})\qquad\qquad 2.\quad G(\lambda\boldsymbol{u})\neq\lambda G(\boldsymbol{u}).

T.ex., följer att

\displaystyle

\begin{align}G(\lambda\boldsymbol{u})&=G\left(\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}\right)=G\left(\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1}\\{\lambda b_1}\\{\lambda c_1}\end{pmatrix}\right)=\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1\cdot\lambda c_1}\\{\lambda^2b_1^2}\\{\lambda b_1+\lambda c_1}\end{pmatrix}\\

&=\lambda\underline{\boldsymbol{e}}\begin{pmatrix}{\lambda a_1c_1}\\{\lambda b_1^2}\\{b_1+c_1}\end{pmatrix}\neq\lambda G(\boldsymbol{u}).\end{align}
Den här artikeln är hämtad från http://wiki.sommarmatte.se/wikis/samverkan/linalg-LIU/index.php/L%C3%B6sning_till_%C3%B6vning_3