"Linear Algebra and its Applications"-chaper6 orthogonality and least squares-orthogonality

In this chapter we discuss the relationship between the vectors defined in the R^n space, which is generally orthogonal, then the orthogonal projection, the best approximation theorem, and so on, these concepts will lay the foundation for the optimal approximate solution of the ax=b of linear equations with no solution.

Orthogonality:

To give the simplest example, in a plane, if the two-dimensional vector's point multiplication is 0, then we can determine that the two vectors are perpendicular to each other, then the two vectors are actually a set of orthogonal vectors on the r^2 vector space.

The following is generalized to the r^n vector space, giving the definition of orthogonality:

Orthogonal set:

Given a set of directional quantities s, when any two elements in s are orthogonal to each other, we call s an orthogonal set.

A concept of a base actually represents some component of all elements in a space (set), and its concept is closely related to linearity, so here we put the implementation on a linear independent.

This theorem actually shows that for the subspace W, with the orthogonal base relative to the advantage of other non-orthogonal bases, it has great convenience in calculating weights, given a vector y in the subspace W, we can make it based on the linear representation of the orthogonal basis.

The proof of the correctness of this calculation is also very good proof that the full use of orthogonal this property can be.

"Linear Algebra and its Applications"-chaper6 orthogonality and least squares-orthogonality