Select all linear transformations that are pirojoctions
A. $T(a, b, c)=(c, a, b)$
B. $T(a, b, c)=(c-b,-a, c)$
C. $T(a, b, c)=(-2 b, b,-a-2 b+c)$
D. $T(a, b, c)=(a, b, c)$

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∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $T(a, b, c) = (c, a, b)$ is a projection, we check if $T(T(a, b, c)) = T(a, b, c)$. Calculating $T(T(a, b, c)) = T(c, a, b) = (b, c, a)$. Since $(b, c, a) eq (c, a, b)$, it is not a projection. ‖ ‖ step 2 ⋮ To determine if $T(a, b, c) = (c - b, -a, c)$ is a projection, we check if $T(T(a, b, c)) = T(a, b, c)$. Calculating $T(T(a, b, c)) = T(c - b, -a, c) = (-a - (c - b), -c + b, c - b)$. Since $(-a - (c - b), -c + b, c - b) eq (c - b, -a, c)$, it is not a projection. ‖ ‖ step 3 ⋮ To determine if $T(a, b, c) = (-2b, b, -a - 2b + c)$ is a projection, we check if $T(T(a, b, c)) = T(a, b, c)$. Calculating $T(T(a, b, c)) = T(-2b, b, -a - 2b + c) = (b, -2b, -a - 2b + c)$. Since $(b, -2b, -a - 2b + c) eq (-2b, b, -a - 2b + c)$, it is not a projection. ‖ ‖ step 4 ⋮ To determine if $T(a, b, c) = (a, b, c)$ is a projection, we check if $T(T(a, b, c)) = T(a, b, c)$. Calculating $T(T(a, b, c)) = T(a, b, c) = (a, b, c)$. Since $(a, b, c) = (a, b, c)$, it is a projection. ‖ ***D*** ∻Key Concept∻ ⚹Projection⚹ ∻Explanation∻ ⚹A projection is a linear transformation $T$ such that $T(T(x)) = T(x)$ for all $x$.⚹◊What is the formula for projecting a vector onto another vector in a three-dimensional space?⍭ Generate me a similar question◊

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