The narrow sense operator actually refers to the mapping from one function space to another function space (or itself).

? As long as the above space is extended to general space, the definition of generalized operator can be vector space. Normed vector space, inner product space or further Banach space and Hilbert space can be used.

In higher algebra, an operator refers to the mapping from one vector space to another, but more attention is paid to its own mapping, that is, transformation.

In higher algebra, projection operator is a linear transformation from vector space to itself, and it is the formalization and popularization of the concept of "parallel projection" in daily life. Projection transformation maps the whole vector space to one of its subspaces, and it is an identity transformation in this subspace.

So the projection operator is a linear transformation of vector space v to itself. p？ Is a projection if and only if p 2 = p

Another definition is more intuitive: p? Is a projection if and only if v has a subspace w, so? p？ Map all elements in v to w, and? P is an identity transformation on w, described in mathematical language, which is:

For example, the vector (x, y,? Z) into a vector (x, y, 0). This is in? x-y？ Projection on the plane. This transformation can be represented by a matrix.

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Because for any vector (x,? y，？ Z), the function of this matrix is:

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Obviously there is p 2 = p, which is a projection operator.