Nibcode Solutions Blog

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.

An LU decomposition (or factorization) of a matrix A is the product of a lower triangular matrix L and an upper triangular matrix U that is equal to A. One of the motivation for an LU decomposition is the fact that this decomposition can be used as an alternative method to solve systems of linear equations, where once the matrix of the system has been decomposed, the solution of the system can be obtained by solving two easy systems, one by the method of forward substitution and the other by the method of backward substitution. The LU decomposition is another approach designed to exploit triangular systems.

Although is very common to be asked to find an LU decomposition for a square matrix, the concepts are extended to rectangular matrices as well. In this problem of the week, you should deal with the LU decomposition for a rectangular matrix.

An important concept related to basis and coordinates is the change of basis matrix. When there are two ordered bases for the same vector space, the change of basis matrix from the first basis to the second one, is the matrix that allows us to get the coordinate vector relative to the second basis by using only the coordinate vector from the first basis, without even knowing the bases themselves.

Understanding the change of basis matrix will help you to understand some problems related to diagonalization and singular value decomposition, among other important concepts which are widely used in many fields of mathematics, physics and engineering.

To solve the problem of this week you will need to use the concepts of coordinate vector and change of basis matrix.

Like last week, this week's problem is also related to spanning sets of vectors, but this time to determine if a vector belongs to a subspace spanned by a set a vectors. Once again, it will be revealed the importance of mastering the basic concepts of rank of a matrix, row operations to convert a matrix to row echelon form and systems of linear equations, to solve problems related to vector spaces.

Vector spaces are one of the key subjects of linear algebra, and their theory has found application in mathematics, engineering, physics, chemistry, biology, the social sciences, and other areas. The theory, basically, consists in generalizing the familiar ideas of geometrical vectors of calculus to vectors of any size, but it provides an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. The beauty of vector spaces theory can be found in every problem, where many of them, are just the appropriate linear-algebraic notion of very well known problems like solving systems of linear equations.

The problem of this week is related to spanning sets and bases, two key concepts from vector spaces you should master. The solution, as usual in most of Linear Algebra problems, uses basic concepts from matrices and their operations.