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[waliant]

\(\displaystyle{ V=Lin\left\{ (1,1,0,0),(0,1,1,0),(0,0,1,1)\right\}}\)
\(\displaystyle{ W=Lin\left\{ (1,0,1,0),(0,2,1,1),(1,2,1,2)\right\}}\)

Ile wynosi \(\displaystyle{ V \cap W}\) ?

[lukasz1804]

\(\displaystyle{ x\in V\iff x=a_V(1,1,0,0)+b_V(0,1,1,0)+c_V(0,0,1,1)=(a_V,a_V+b_V,b_V+c_V,c_V)}\) dla pewnych \(\displaystyle{ a_V,b_V,c_V\in\RR}\)
\(\displaystyle{ x\in W\iff x=a_W(1,0,1,0)+b_W(0,2,1,1)+c_W(1,2,1,2)=(a_W+c_W,2b_W+2c_W,a_W+b_W+c_W,b_W+2c_W)}\) dla pewnych \(\displaystyle{ a_W,b_W,c_W\in\RR}\)

Stąd mamy \(\displaystyle{ (a_V,a_V+b_V,b_V+c_V,c_V)=(a_W+c_W,2b_W+2c_W,a_W+b_W+c_W,b_W+2c_W)}\), czyli \(\displaystyle{ \begin{cases} a_V&=a_W+c_W \\ a_V+b_V&=2b_W+2c_W \\ b_V+c_V&=a_W+b_W+c_W \\ c_V&=b_W+2c_W \end{cases}\iff\begin{cases} a_V=a_W+c_W \\ b_V=-a_W+2b_W \\ c_V=2a_W-b_W+c_W \\ c_V=b_W+2c_W \end{cases}\iff\begin{cases} a_V=a_W+c_W \\ b_V=-a_W+2b_W \\ c_V=2a_W-b_W+c_W \\ c_W=2a_W-2b_W \end{cases}\implies\begin{cases} a_V=3a_W-2b_W \\ b_V=-a_W+2b_W \\ c_V=4a_W-3b_W \end{cases}}\)

To daje, że \(\displaystyle{ x\in V\cap W\iff x=(a_V,a_V+b_V,b_V+c_V,c_V)=(3a_W-2b_W,4a_W,3a_W-b_W,4a_W-3b_W)=a_W(3,4,3,4)-b_W(2,0,1,3)}\), więc

\(\displaystyle{ V\cap W=\mbox{lin}\big\{(3,4,3,4),(2,0,1,3)\big\}}\).