Tips och lösning till U 13.13d
SamverkanLinalgLIU
Tips 1
Hej 1
Tips 2
Hej 2
Tips 3
Hej 3
Lösning
Enligt Sats 12.22 ges varje vektor entydigt av
\boldsymbol{u}=\underbrace{(\boldsymbol{u}|\boldsymbol{e}_1)\boldsymbol{e}_1+(\boldsymbol{u}|\boldsymbol{e}_2)\boldsymbol{e}_2+\boldsymbol{u}|\boldsymbol{e}_3)\boldsymbol{e}_3} _{P_W(\boldsymbol{u})=\boldsymbol{u}_{\parallel W}} \quad + \underbrace{ (\boldsymbol{u}|\boldsymbol{e}_4)\boldsymbol{e}_4}_{P_{W^\perp}(\boldsymbol{u})=\boldsymbol{u}_{\parallel W^\perp}}.
Om \displaystyle \boldsymbol{u}=(2,2,6,2)^t , så är
P_{W}(\boldsymbol{u}) = \boldsymbol{u}_{\parallel W} = (\boldsymbol{u}|\boldsymbol{e}_1)\boldsymbol{e}_1+(\boldsymbol{u}|\boldsymbol{e}_2)\boldsymbol{e}_2 +(\boldsymbol{u}|\boldsymbol{e}_3)\boldsymbol{e}_3 =\frac{1}{5} (14,18,22,6)^t
och
P_{W^\perp}(\boldsymbol{u}) = \boldsymbol{u}_{\perp W} = (\boldsymbol{u}|\boldsymbol{e}_4)\boldsymbol{e}_4= \frac{1}{5}(-4,-8,8,4)^t.
