Mathematics, 08.09.2020 14:01 AgentPangolin
Find the closest point to y in the subspace W spanned by v1 and v where y = [ 3 9 5 -8], v1 = [5 4 3 2], v2 = [-4 5 -2 3]
Answers: 2
![\bold{ \left[\begin{array}{c}\frac{127}{27} \\ \ &\frac{65}{18} \\ \ &\frac{76}{27}\\ \ &\frac{97}{54}\\\ \end{array}\right]}](/tpl/images/0745/7721/069d8.png)
![v_1=\left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \\](/tpl/images/0745/7721/9f390.png)
![v_2=\left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] \\](/tpl/images/0745/7721/e4b06.png)
:![= \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]](/tpl/images/0745/7721/7a92f.png)
" are orthogonal vectors that is: 
from the above orthogonal basis of subspace w and the shortest distanc value of vector y:![=\left[\begin{array}{c}3&9&-5&8\\\ \end{array}\right] \\\\](/tpl/images/0745/7721/d9107.png)
![= \frac{\left[\begin{array}{c}3&9&-5&8\\\ \end{array}\right] \cdot \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] }{\left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \cdot \left[\begin{array}{c}5&4&3&2\\ \end{array}\right]} \times \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] +](/tpl/images/0745/7721/4f15b.png)
![\frac{\left[\begin{array}{c}3&9&5&-8\\\ \end{array}\right] \cdot \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] }{\left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] \cdot \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]} \times \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]](/tpl/images/0745/7721/1df56.png)
![= \frac{50}{54} \left[\begin{array}{c}5&4&3&2\\\ \end{array}\right] - \frac{1}{54} \left[\begin{array}{c}-4&5&-2&3\\\ \end{array}\right]\\\\](/tpl/images/0745/7721/1402e.png)
![= \left[\begin{array}{c}\frac{127}{27} \\ \ &\frac{65}{18} \\ \ &\frac{76}{27}\\ \ &\frac{97}{54}\\\ \end{array}\right]](/tpl/images/0745/7721/70499.png)
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Find the closest point to y in the subspace W spanned by v1 and v
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