slides pptx - Tennessee State University
... What is a Vector Space? A vector space is a set of objects that may be added together or multiplied by numbers (called scalars) Scalars are typically real numbers But can be complex numbers, rational numbers, or generally any field ...
... What is a Vector Space? A vector space is a set of objects that may be added together or multiplied by numbers (called scalars) Scalars are typically real numbers But can be complex numbers, rational numbers, or generally any field ...
mathematics 217 notes
... where Xi ⊂ W ⊂ V are T -invariant subspaces and T |Xi : Xi −→ Xi is represented by the matrix B(ni ; ai ). Each subspace Xi contains a single eigenvector vi with eigenvalue ai , by the preceding theorem. Let K ⊂ V be the kernel of the linear transformation T , so dim K = n − m, and suppose that dim( ...
... where Xi ⊂ W ⊂ V are T -invariant subspaces and T |Xi : Xi −→ Xi is represented by the matrix B(ni ; ai ). Each subspace Xi contains a single eigenvector vi with eigenvalue ai , by the preceding theorem. Let K ⊂ V be the kernel of the linear transformation T , so dim K = n − m, and suppose that dim( ...
Recitation Notes Spring 16, 21-241: Matrices and Linear Transformations February 9, 2016
... (b) Let {Am }m∈N be a sequence of n × n upper triangular matrices. Prove by induction that A1 A2 · · · An is upper triangular for all n ∈ N. Solution. We shall prove by induction on n. Base cases: When n = 1, this is trivially true. Induction hypothesis: Let n ∈ N. Assume that A1 A2 · · · An is upp ...
... (b) Let {Am }m∈N be a sequence of n × n upper triangular matrices. Prove by induction that A1 A2 · · · An is upper triangular for all n ∈ N. Solution. We shall prove by induction on n. Base cases: When n = 1, this is trivially true. Induction hypothesis: Let n ∈ N. Assume that A1 A2 · · · An is upp ...
Linear Algebra - John Abbott Home Page
... the field of Social Science such as production problems (systems of linear equations and linear combinations), Leontief Input-Output Model (systems of linear equations and the inverse of a matrix) and the optimization of (economic) functions (vector spaces and the Simplex method). In this way, the b ...
... the field of Social Science such as production problems (systems of linear equations and linear combinations), Leontief Input-Output Model (systems of linear equations and the inverse of a matrix) and the optimization of (economic) functions (vector spaces and the Simplex method). In this way, the b ...
