5. Consider a vector subspace whose basis vectors are $\begin{bmatrix} 2\\1\\2 \end{bmatrix}$ and $\begin{bmatrix} -2\\2\\1 \end{bmatrix}$. What is the orthogonal projection of the vector $\begin{bmatrix} 1\\2\\-2 \end{bmatrix}$ onto this subspace? A. $\begin{bmatrix} 0\\0\\0 \end{bmatrix}$ B. $\begin{bmatrix} 1\\2\\-2 \end{bmatrix}$ C. $\begin{bmatrix} 0\\0\\0 \end{bmatrix}$ D. $\frac{1}{9}\begin{bmatrix} 2\\1\\2 \end{bmatrix}$
Added by Todd S.
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To do this, we can use the formula for the orthogonal projection: proj_v(u) = (u · v) / (v · v) * v where u is the vector we want to project, and v is a basis vector of the subspace. In this case, u = 2 and v = 1. So we have: proj_1(2) = (2 · 1) / (1 · 1) * 1 Show more…
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