Final answer:
Orthonormal vectors produce new vectors with the same length but different directions. A vector can belong to multiple subspaces without being the zero vector. The orthogonal projection of a vector onto a subspace is the closest vector in the subspace. The best approximation of a vector by elements of a subspace is the vector resulting from projecting the vector onto the subspace.
Step-by-step explanation:
In linear algebra, vectors with orthonormal columns are called orthonormal vectors. For any vector x and orthonormal matrix Q, the expression QTx yields a vector y with the same length as x but a different direction. This means that for any vector x and orthonormal matrix Q, the resulting vector y is in a different direction from x.
If a subspace V is a subset of a vector space U, and a vector v is in both V and U, then it must also be in the intersection of V and U. Therefore, the vector v is not necessarily the zero vector, but it belongs to both subspaces.
The orthogonal projection of a vector x onto a subspace V is a vector p that is the closest vector to x in V. The orthogonal projection vector p must be in the subspace V, but it is not necessarily equal to the vector x.
In the orthogonal decomposition theorem, each term in the decomposition is itself an orthogonal projection of the vector x onto a subspace of the original vector space. In other words, the vector x is decomposed into a sum of orthogonal projection vectors.
The best approximation to a vector x by elements of a subspace is given by the vector that results from projecting x onto the subspace. This vector represents the closest vector in the subspace to x.