By Kalkbrener M.
In removing conception structures of algebraic equations in different variables are studied so that it will manage stipulations for his or her solvability in addition to formulation for calculating their options. during this Ph.D. thesis we're taken with the appliance of recognized algorithms from removing idea lo difficulties in geometric modeling and with the advance of recent equipment for fixing structures of algebraic equations.
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U∗ii , U∗i+1,i , . . , U∗ni , . . , U∗nn spans the linear space of all n × n symmetric matrices. Moreover, this set is linearly independent (as can be easily verified). Thus, the set {U∗11 , U∗21 , . . , U∗n1 , . . , U∗ii , U∗i+1,i , . . , U∗ni , . . , U∗nn } is a basis for the linear space of all n × n symmetric matrices. c. Existence of a basis It was determined in Subsection b that the linear space Rm×n has a basis and that the linear space of all n×n symmetric matrices has a basis. Does every linear space (of m × n matrices) have a basis?
Let A1 , . . , Ak represent the matrices in S. Define S ∗ to be the subset of S obtained by successively applying to each of the matrices A1 , . . , Ak the following algorithm: include the matrix as a member of S ∗ if it is nonnull and if it is not expressible as a linear combination of any matrices already included in S ∗ . 2 that the subset S ∗ formed in this way is linearly independent. 1) that S ∗ spans V. We conclude that S ∗ is a basis for V. D. 11 can be used to construct a basis for V and, in the process, to determine the dimension of V.
0 0 Arr then A is called an upper block-triangular matrix. Similarly, if Aij i 1, . . , r, that is, if 0 ··· 0 A11 A21 A22 0 A . , . .. .. Ar1 Ar2 Arr 0 for j > then A is called a lower block-triangular matrix. To indicate that A is either upper or lower block-triangular (without being more specific), A is referred to simply as block-triangular. 2) is block-diagonal if and only if it is both upper block-triangular and lower block-triangular. Note also that, if m n r (in which case each block of A consists of a single element), saying that A is block diagonal or upper or lower block triangular is equivalent to saying that A is diagonal or upper or lower triangular.