3 5 2 2 3 1 3x+5y=2 2x+3y=1 replace with
... We can interpret an SLE in terms of (column) vectors; writing vi = ith column of the coefficient matrix, and b=the column of target values, then our SLE really reads as a single equation x1 v1 + · · · + xn vn = b. The lefthand side of this equation is a linear combination of the vectors v1 , . . . , ...
... We can interpret an SLE in terms of (column) vectors; writing vi = ith column of the coefficient matrix, and b=the column of target values, then our SLE really reads as a single equation x1 v1 + · · · + xn vn = b. The lefthand side of this equation is a linear combination of the vectors v1 , . . . , ...
Condensation Method for Evaluating Determinants
... This method, while perfectly reliable, can get very long and laborious as the dimensions of the determinant increase. These days, with the help of packages like Mathematica, Maple, or Matlab, such intricate calculations can be done in a matter of seconds. But in the 19th century, of course, they had ...
... This method, while perfectly reliable, can get very long and laborious as the dimensions of the determinant increase. These days, with the help of packages like Mathematica, Maple, or Matlab, such intricate calculations can be done in a matter of seconds. But in the 19th century, of course, they had ...
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... Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. ...
... Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. ...
n-Dimensional Euclidean Space and Matrices
... the dot product in three space are still valid for the dot product in Rn . Each can be proven by a simple computation. Vector Spaces and Inner Product Spaces. The notion of a vector space focusses on having a set of objects called vectors that one can add and multiply by scalars, where these operati ...
... the dot product in three space are still valid for the dot product in Rn . Each can be proven by a simple computation. Vector Spaces and Inner Product Spaces. The notion of a vector space focusses on having a set of objects called vectors that one can add and multiply by scalars, where these operati ...
Rank (in linear algebra)
... The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A). Alternatively and equivalently, we can define the rank of A as the dimen ...
... The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A). Alternatively and equivalently, we can define the rank of A as the dimen ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.