Rank Theorem (Rank-Nullity Thm), 4 Fundamental Subspaces, Invertible Matrix Theorem, Change of Basis

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(a.k.a. Differential Equations with Linear Algebra, Lecture 20B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 20B).

#vectorspace #ranktheorem #invertiblematrixtheorem

(0:00) Applications of abstract vector spaces to differential equations and difference equations are soon to come.
(0:56) Dimension of a vector space
(3:12) Basis Theorem
(5:49) Row space Row(A) of an m x n matrix A
(6:47) Transpose A^T of A (which is n x m when A is m x n)
(8:44) Row equivalent matrices have equal row spaces. In addition, the nonzero rows of an echelon form of A form a basis for Row(A).
(9:41) Rank Theorem (a.k.a. Rank-Nullity Theorem)
(11:50) Four Fundamental Subspaces associated with A (an m x n matrix)
(13:18) Orthogonality relations and visualization
(16:23) Invertible Matrix Theorem
(18:50) Change of Coordinates Matrix in Euclidean Space
(24:08) Change of Basis in General (for abstract finite-dimensional vector spaces)
(29:20) Changes of coordinates matrix in the general case where dim(V) = n
(32:08) Row reduction algorithm for computing the change of coordinates matrix P when V = R^n.
(33:25) Example when V = R^2.

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  1. Bill Kinney
  2. rank theorem
  3. rank theorem linear algebra
  4. rank nullity
  5. rank nullity theorem
  6. four fundamental subspaces linear algebra
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Автор

Professor Kinney, thank you for a basic analysis and understanding of the Rank Theorem, 4 Fundamental Subspaces, the Invertible Matrix Theorem and the Change of Basis in Linear Algebra. The continuous emphasis on theory is important, however there is a linked between Differential Equations and Linear Algebra with change of basis matrix and the transpose of a matrix for different systems.

georgesadler