User:Srilalithkumar/HW2

$$\left ( 1 \right ) \overrightarrow{b_{i}}\cdot P\left (\overrightarrow{v}  \right ) = 0;$$ Where $$i\in $$ $$ \left \{ 1,2,3,.....n \right \}$$ $$\left ( 2 \right ) \overrightarrow w \cdot P\left (\overrightarrow{v}  \right ) = 0;$$      $$ \forall \left\{ \alpha_{1},\alpha_{2},\alpha_{3}........\alpha_{n}\right\}$$$$\in\mathbb{R}^n$$; Where $$\overrightarrow w = \sum_{i=1}^{n}\alpha_{i} \cdot \overrightarrow{b_{i}} $$ Show that $$\left ( 1 \right ) \Leftrightarrow \left ( 2 \right )$$
 * solution: :

Multiply both sides of the equation $$\left ( 1 \right)$$ by $$ \alpha_{i} $$ $$\left ( 3 \right )\alpha_{i}\cdot\overrightarrow{b_{i}}\cdot P\left (\overrightarrow{v} \right ) = \alpha_{i}\cdot 0;$$ $$\left ( 4 \right )\alpha_{i}\cdot\overrightarrow{b_{i}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$

when $$i=1$$ in Equation 3,we get $$\left ( 4.1 \right )\alpha_{1}\cdot\overrightarrow{b_{1}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$ when $$i=2$$ in Equation 3,we get $$\left ( 4.2 \right )\alpha_{2}\cdot\overrightarrow{b_{2}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$ when $$i=3$$ in Equation 3,we get $$\left ( 4.3 \right )\alpha_{3}\cdot\overrightarrow{b_{3}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$ when $$i=4$$ in Equation 3,we get $$\left ( 4.4 \right )\alpha_{4}\cdot\overrightarrow{b_{4}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$ . . . . when $$i=n$$ in Equation 3,we get $$\left ( 4.n \right )\alpha_{n}\cdot\overrightarrow{b_{n}}\cdot P\left (\overrightarrow{v} \right ) = 0;$$

Summation of equations 4.1 to 4.n can be represented by $$\sum_{i=1}^{n}\alpha_{i}\cdot b{i}\cdot P\left (\overrightarrow{v} \right ) = 0$$

and substituting $$\overrightarrow w = \sum_{i=1}^{n}\alpha_{i} \cdot \overrightarrow{b_{i}} $$ We get $$\left ( 5 \right )\overrightarrow w \cdot P\left (\overrightarrow{v} \right ) = 0;$$ Which is equal to Equation $$\left ( 2 \right )$$