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A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.
The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent. [2] A linear code of length n transmits blocks containing n symbols. For example, the [7,4,3] Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct codewords differ ...
Since the tensor product of linear maps is itself a linear map, [2] and because every tensor admits a tensor rank decomposition, [1] the above expression extends linearly to all tensors. That is, for a general tensor A ∈ V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d {\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}} , the ...
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that () = =, while, by convention, 0-forms are defined to be scalars: () = =. The determinant on n × n {\displaystyle n\times n} matrices, viewed as an n {\displaystyle n} argument function of the column vectors, is an important example of an alternating ...
Multi-objective linear programming is a subarea of mathematical optimization. A multiple objective linear program (MOLP) is a linear program with more than one objective function. An MOLP is a special case of a vector linear program. Multi-objective linear programming is also a subarea of Multi-objective optimization.
The people in Donald Trump’s orbit are floating some dramatic ideas that would remake the way banks are regulated. There are lots of questions about whether any of the ideas will come to pass.
The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values. The determinant , permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns).