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SCx.y.z)=( 3x, 3y.3=) Do S and T commute? TCXxy.z)= (x.y > %3D

SCx.y.z)=( 3x, 3y.3=) Do S and T commute? TCXxy.z)= (x.y > %3D

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The image contains handwritten equations exploring whether two functions, S and T, commute. The question posed is: "Do S and T commute?" Here are the definitions of the functions: 1. **Function S**: - \( S(x, y, z) = (3x, 3y, 3z) \) - This function scales each component of the input vector \((x, y, z)\) by 3. 2. **Function T**: - \( T(x, y, z) = (x, y) \) - This function projects the input vector \((x, y, z)\) onto the xy-plane, effectively eliminating the z-component. To determine if S and T commute, we would need to check if applying S and then T yields the same result as applying T and then S, given the same input. In other words, if \( T(S(x, y, z)) = S(T(x, y, z)) \) for all input vectors \((x, y, z)\).
Transcribed Image Text:The image contains handwritten equations exploring whether two functions, S and T, commute. The question posed is: "Do S and T commute?" Here are the definitions of the functions: 1. **Function S**: - \( S(x, y, z) = (3x, 3y, 3z) \) - This function scales each component of the input vector \((x, y, z)\) by 3. 2. **Function T**: - \( T(x, y, z) = (x, y) \) - This function projects the input vector \((x, y, z)\) onto the xy-plane, effectively eliminating the z-component. To determine if S and T commute, we would need to check if applying S and then T yields the same result as applying T and then S, given the same input. In other words, if \( T(S(x, y, z)) = S(T(x, y, z)) \) for all input vectors \((x, y, z)\).
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