"There are many types of projections that could be used geometrically. orthogonal projections are just one possibility. Why do we often prefer orthogonal projections? I've found out someone's answer but it is hard to understand. Could you explain more to understand?"
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Akash M.
In $R^{3}$ the orthogonal projections onto the $x$ -axis, $y$ -axis, and $z$ -axis are $$\begin{array}{c}T_{1}(x, y, z)=(x, 0,0), \quad T_{2}(x, y, z)=(0, y, 0) \\T_{3}(x, y, z)=(0,0, z)\end{array}$$ respectively. (a) Show that the orthogonal projections onto the coordinate axes are matrix operators, and then find their standard matrices. (b) Show that if $T: R^{3} \rightarrow R^{3}$ is an orthogonal projection onto one of the coordinate axes, then for every vector $x$ in $R^{3}$, the vectors $T(\mathbf{x})$ and $\mathbf{x}-T(\mathbf{x})$ are orthogonal. (c) Make a sketch showing $\mathbf{x}$ and $\mathbf{x}-T(\mathbf{x})$ in the case where $T$ is the orthogonal projection onto the $x$ -axis.
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Basic Matrix Transformations in $R^{2}$ and $R^{3}$
For each orthogonal projection operator in Table 4 use the standard matrix to compute $T(1,2,3),$ and convince yourself that your result makes sense geometrically.
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