Projection and unitary change are different operations
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A basis change preserves the whole vector. Projection keeps only one orthogonal component. Let \(e_s=c_{12}/\sqrt{48}\) and \(e_r=(I-P)x/\sqrt{102}\). They are orthonormal, so the normalized input has the exact two-dimensional representation
\[
\frac{x}{\sqrt{150}}=\sqrt{\frac8{25}}e_s+\sqrt{\frac{17}{25}}e_r.
\]
The Fourier transform preserves the input norm and permits exact inversion. The projector maps every complementary vector to zero. That distinction controls what may legitimately be called reconstruction, filtering, or measurement.