Question

Wite **the sum of two vectons, one in Span {u) and one in Span (wa). Assume that (.....) is an orthogonal besis Type an integer or simplified traction for each max element) Verity that {.uz) is an orthogonal sot, and then find the orthogonal projection of y onto Span(uz) y To verty that (0-uz) as an orthogonal set, find u, uz 2-0 (Simplify your answer.) The projection of yonte Span (0,2) 0 (Simplify your answers.) LetW be the subspace spanned by u, and u. Wrte y as the sum of a vector in W and a vector orthogonal to W. y- The sum is y=y+z, where = is in Wandz= 1 orthogonal to W. (Simplify your answers.) Find the closest point to y in the subspace Wspanned by V, and V. ys The closest point to y in W is the vector DOOD (Simplify your answers)

Answer 1

1)

By forming Matrix with u1, u2, u3, u4 as columns and

AX =B where A = [u1, u2, u3, u4 ], X = [x1, x2, x3, x4]^{T ,} , B = [x]

fining the Solution for the System of Linear Equations we get

- 1 15 -6 5 11 1 X = A-?B= 0 7 1 9 90 -7 -8 1 1 -1 -1 1-8 1 3 0

so we have $x = \frac{1}{22}((-10u_1+9u_2+6u_3)+(29u_4))$

2)

Recall that the vector projection of a vector ū`onto another vector Tºis given by proj, (u u. ||0 w The projection of ū onto a plane can be calculated by subtracting the component of ū that is orthogonal to the plane from u . If you think of the plane as being horizontal, this means computing ū minus the vertical component of ū leaving the horizontal component. This "vertical" component is calculated as the projection of ū onto the plane normal vector n. projplane (W) = ū – projna(w) un n = 2 n

Normal Vector = $\begin{pmatrix}2&3&0\end{pmatrix}\times \begin{pmatrix}-3&2&0\end{pmatrix}=\begin{pmatrix}0&0&13\end{pmatrix}$

$proj(y)_{span(u_1, u_2)} = \overrightarrow{y}- \frac{\overrightarrow{y}. \overrightarrow{n}}{||\overrightarrow{n}||^2}.\overrightarrow{n} =(4,5,-1)-\frac{(4,5,-1).(0,0,13)}{13^2}(0,0,13) = (4,5,-1)+(0,0,1)= (4,5,0)$

3)

Similar to above method we can find the Prjection of y(-1,5,4) in W as $=(-1,5,4)-\frac{(-1,5,4).(-8,1,7)}{114}(-8,1,7) = (-1,5,4)-\frac{41}{114}(-8,1,7)$

And the Orthogonal Projection to W is   $\frac{41}{114}(-8,1,7)$.

Summing it up we get y.

Please Ask for any futher clarifications!!!