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Projections are mathematical operations that map elements from a space onto a subspace or onto another target space, often while preserving some structure. In linear algebra, a projection is a linear operator P on a vector space V such that P^2 = P. The image of P is the projected subspace; the kernel is the set mapped to zero. Projections are idempotent; applying the projection twice yields the same result as applying it once. If P is orthogonal with respect to an inner product, it maps each vector to the closest vector in the target subspace, and the space decomposes into the direct sum of the subspace and its orthogonal complement.

Typical examples include projecting vectors in R^n onto a coordinate axis (e.g., projecting onto the x-axis), or onto a plane or line spanned by a given set of vectors. The general formula for an orthogonal projection onto a subspace W with an orthonormal basis {u_i} is P = sum_i u_i u_i^T; for projection onto the column space of a full-rank matrix A, P = A(A^T A)^{-1}A^T.

Map projections in cartography are different uses of the word. A map projection transforms the curved surface of the Earth (a sphere or ellipsoid) to a plane, inevitably distorting certain properties such as area, shape, distance, or direction. Different projections preserve or approximate different properties according to the purpose, leading to many well-known choices, like Mercator for navigation or equal-area projections for statistical maps.

Projections also occur in data analysis as dimensionality reduction: projecting high-dimensional data into a lower-dimensional space for visualization or processing. Techniques include principal component analysis, random projections, and multidimensional scaling; these reduce dimensionality while attempting to preserve variance or pairwise distances.

In forecasting and planning, projections are estimates of future values produced by models. They are contingent on assumptions and inputs and are often presented as scenarios to illustrate uncertainty and potential ranges, rather than as precise predictions.