Problem of the week - Gram-Schmidt process
February 6, 2019
Orthogonal bases have some practical advantages and are very useful when dealing with projections onto subspaces. These bases are defined in spaces equipped with an inner product also called a dot product, and by definition, a basis is called orthogonal if every pair of basis vectors are orthogonal, that is, their inner product is 0. When the length of each vector is 1 (vectors are normalized), the basis is called an orthonormal basis.
In an inner product space, it is always possible to get an orthonormal basis starting from any basis, by using the Gram-Schmidt algorithm. To solve this problem of the week you will need to prove you master the Gram–Schmidt process, and you also need to compute the change of basis matrix.
Problem of the week - Orthogonal bases
January 16, 2019
Many concepts of Linear Algebra have emerged from geometric problems, and then generalized to non-visual higher-dimensional spaces. Some of the most widely used geometric concepts are length, distance and perpendicularity, which provide powerful geometric tools for solving many applied problems, including the least-squares problems.
These all three notions are defined in terms of the inner product of two vectors, which is also the key concept to deal with orthogonal bases, the subject of the problem of this week. Orthogonal bases, and particularly orthonormal bases, are very useful when dealing with projections onto subspaces, among other problems.
