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Applications
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
where ⊕ is the direct sum operator. The complementary space Wj is related to its one-dimensional cousin. This relationship is stated in the following lemma.
Bases and Dimension
Published in Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, 2017
Ravi P. Agarwal, Cristina Flaut
The vector space V is said to be a direct sum of its subspaces U and W, denoted as U ⊕ W, if for every v ∊ V there exist unique vectors u ∊ U and w ∊ W such that v = u + w.
Nonassociative Algebras
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Murray R. Bremner, Lucia I. Murakami, Ivan P. Shestakov
Given an algebra A with two ideals B and C, we say that A is the (internal) direct sum of B and C if A = B ⊕ C (direct sum of subspaces).
Estimation method for inverse problems with linear forward operator and its application to magnetization estimation from magnetic force microscopy images using deep learning
Published in Inverse Problems in Science and Engineering, 2021
For a moment vector ( denotes the transpose) defined in Section 2.2 and define by Consequently, the ith component of is equal to the ith component of μ if the component is the magnetic moment of one of the ℓth cells vertically from the top of M, and otherwise, the ith component of is 0. For example, the non-zero components of correspond to the cells of the top surface of M. (Furthermore, see Example 2.3.) We define and we call the ℓth piece of In particular, and are called the top piece and the bottom piece, respectively. Furthermore, with a small value of ℓ is called an upper piece while with a large value of ℓ is called a lower piece. Then, we have There exists a natural projection where the symbol ⊕ denotes the direct sum of vector spaces or their subsets. However, does not hold because μ does not have zero components and has zero components.
New results on algorithms for the computation of output-nulling and input-containing subspaces
Published in International Journal of Control, 2023
Lorenzo Ntogramatzidis, Fabrizio Padula, Augusto Ferrante
Notation: Throughout this paper, the symbol will stand for the origin of any vector space. For convenience, a linear mapping between finite-dimensional spaces and a matrix representation with respect to a particular basis are not distinguished notationally. The image and the kernel of matrix A are denoted by and , respectively. When A is square, we denote by the spectrum of A. Given a linear map and a subspace of , the symbol stands for the inverse image of under the linear map A, i.e. . If , the restriction of the map A to is denoted by . If and is A-invariant, the eigenvalues of A restricted to are denoted by . If and are A-invariant subspaces and , the mapping induced by A on the quotient space is denoted by , and its spectrum is denoted by . The symbol ⊕ stands for the direct sum of subspaces. The symbol denotes union with any common elements repeated.