Question

Let W be the subspace spanned by ui and u2, and write y as the sum of a vector vi in Wand a vector v2 orthogonal to w -4 -8 NOTE: You should fill in all the boxes below before submitting. Both vectors are to be submitted at once. Answers can be entered as numerical formulae, or rounded to 3 decimal places. You may use a calculator for the arithmetic operations

Answer 1

Solution:

Write   _{$\small y=v_{1}+v_{2}$} , where _{$\small v_{1}=proj_{W}\left ( y \right )$} is the projection of $\small y$ on $\small W$ , and   _{$\small v_{2}=y-v_{1}$} is orthogonal to $\small W.$   

Since   _{$\small u_{1}\cdot u_{2}=0$} , calculate   _{$\small v_{1}$} using the formula

$\small v_{1}=\left ( \frac{y\cdot u_{1}}{u_{1}\cdot u_{1}} \right )u_{1}+\left ( \frac{y\cdot u_{2}}{u_{2}\cdot u_{2}} \right )u_{2}$

   _{$\small y\cdot u_{1}=-12-25-16=-53$} ,

$\small y\cdot u_{2}=4+15-72=-53$ ,

$\small u_{1}\cdot u_{1}=9+25+4=38$ ,

$\small u_{2}\cdot u_{2}=1+9+81=91$

$\small \therefore v_{1}=\left ( \frac{-53}{38} \right )u_{1}+\left ( \frac{-53}{91} \right )u_{2}$

$\small \therefore v_{1}=\begin{bmatrix} \frac{-12455}{3458}\\ \\ \frac{18073}{3458}\\ \\ \frac{-13886}{1729}\\ \\ \end{bmatrix}\in W$

$\small \therefore v_{2}=y-v_{1}=\begin{bmatrix} -4\\ 5\\ -8\\ \end{bmatrix}-\begin{bmatrix} \frac{-12455}{3458}\\ \\ \frac{18073}{3458}\\ \\ \frac{-13886}{1729}\\ \\ \end{bmatrix}=\begin{bmatrix} \frac{-1377}{3458}\\\\ \frac{-783}{3458}\\ \\ \frac{54}{1729}\\ \end{bmatrix}$

$\small v_{1}=\begin{bmatrix} \frac{-12455}{3458}\\ \\ \frac{18073}{3458}\\ \\ \frac{-13886}{1729}\\ \\ \end{bmatrix}\: \: \: \: \: \: \: \: \: \small v_{2}=\begin{bmatrix} \frac{-1377}{3458}\\\\ \frac{-783}{3458}\\ \\ \frac{54}{1729}\\ \end{bmatrix}$