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Linear Algebra

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Moderate

Image of an Orthogonal Projection

LINALG-2BSAPR

Consider the orthogonal projection of $\mathbb{R}^3$ onto a subspace $S$ of $\mathbb{R}^3$. Suppose that the image of the vector $u_1=\begin{bmatrix} 1\\\ 1 \\\ 1\end{bmatrix}$ is the vector $v_1= \frac{2}{3}\begin{bmatrix} 1\\\ 1 \\\ 2\end{bmatrix},$ and that the image of the vector $u_2=\begin{bmatrix} 3\\\ 2 \\\ 1\end{bmatrix}$ is the vector $v_2= \frac{1}{3}\begin{bmatrix} 5\\\ 2 \\\ 7\end{bmatrix}.$

Which of the following is the equation of the subpace $S$?

A

$2x+3y-z=0$

B

$-x+y+2z=0$

C

$x+y-z=0$

D

$-3x+2y-5z=0$