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Introduction and Background
Published in Ossama Abdelkhalik, Algorithms for Variable-Size Optimization, 2021
Two vectors are said to be orthogonal if they are perpendicular to each other; that is is the scalar product of the two vectors is zero. The vectors {x→1,x→2,⋯,x→m} are said to be mutually orthogonal if every pair of vectors is orthogonal. A set of vectors is said to be orthonormal if every vector is of unit magnitude and the set of vectors are mutually orthogonal. A set of vectors that are orthogonal are also linearly independent.
Analysis and Processing in the Frequency Domain
Published in Yevgeniy V. Galperin, An Image Processing Tour of College Mathematics, 2021
So, we see that orthogonal bases are very convenient! It is easy (conceptually and computationally) to calculate the coordinates of a vector with respect to an orthogonal basis. But is every orthogonal set of vectors a basis for its span? We recall that in order for a given set of vectors to form a basis, it must first and foremost be linearly independent. Here too, orthogonality proves to be a very useful property. Whereas the general problem of determining whether a given set of vectors is linearly independent is equivalent to solving a certain linear system (by means of row-reduction) and establishing that it only has the trivial solution, any orthogonal set of vectors is automatically linearly independent! Indeed, suppose that the nonzero vectors {vk} are pairwise-orthogonal and that ∑kckvk=0
Numerical Solution of Linear Algebraic Equations
Published in Jan Ogrodzki, Circuit Simulation Methods and Algorithms, 2018
The concept of the linear combination α1x1+α2x2+…+αixi. of a number of vectors, where α1,…αi are real weights, is of great importance in linear algebra. Certain vectors are known as linearly independent if none of them can be expressed as a linear combination of the others. They are linearly dependent otherwise. The linear mapping A may have linearly independent columns or some of them may be linear combinations of the others. A maximum number of linearly independent rows and columns of the matrix is known as its order, namely ord(A). Among n vectors of a dimension m, at most m and obviously not more then n are linearly dependent. Hence ord(A)≤ min (m, n). Notice that if the system (3.1) has a solution then the right-hand side (RHS) b is a linear combination of columns of A. Hence, the number of linearly independent columns of A is the same as the number of linearly independent columns of the matrix A│b (A extended with b) only when b is a linear combination of columns of A (that is when a solution of the equations exists). This leads to the Kronecker-Cappelli Theorem:
Sparsity constrained optimization problems via disjunctive programming
Published in Optimization, 2022
N. Movahedian, S. Nobakhtian, M. Sarabadan
Consider the following sparsity constrained problem: An easy calculation shows that . The unique optimal solution is and . It is clear that the vectors are linearly dependent. Thus does not satisfy SCO-MFCQ.
Applying combinatorics in the design of multiversion exams
Published in International Journal of Mathematical Education in Science and Technology, 2023
Associate with the subspace H the matrix . The set of columns of A is where is the standard unit vector. Note that deleting the column from A gives the matrix whose rows are the vectors associated with G. We wish to show that any three vectors in are linearly independent for any field , and any three vectors in C are linearly independent if is of characteristic 2. Clearly, any set of three vectors in C including at least two of the is linearly independent. We consider the other cases. Note that a set of three vectors is linearly independent if and only if the matrix M with these vectors as columns has nonzero determinant. for distinct . The determinant of M is . for distinct . The determinant of M is . for distinct . The determinant of M is . This is 0 if and only if . If is of characteristic 2, then if and only if . So for distinct nonzero if of characteristic 2, the determinant of M is nonzero. for distinct . The determinant of M (a Vandermonde matrix) is .