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I want to verify or rather show that a given matrix ##T## does represent a projection from ##\mathbb{R^{3}}## to a particular plane, also lying in ##\mathbb{R^{3}}##. Would it be enough to pre-multiply that matrix to an arbitrary vector ##(x,y,z)##, and see if the resulting vector is orthogonal to the normal vector of that given plane, thus implying that the vector is projected successfully to the plane?

Or do I need to row reduce the matrix ##T## until I can see the basis vectors used in the original ##T##, and verify that they all lie on the plane? Or rather since I can also get the basis of the kernel, will showing that the basis of the kernel is parallel with the normal of the plane enough? Geometrically I imagine that the kernel space is all of the vectors that are orthogonal to the plane and their projection to that plane will be 0.

Thanks

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# Verifying that a matrix T represents a projection operation

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