Linear Transformations Defined by Matrix/Vector Multiplication (Linear Combinations of Columns) скачать в хорошем качестве

Linear Transformations Defined by Matrix/Vector Multiplication (Linear Combinations of Columns)

Differential Equations and Linear Algebra Lecture 7B. Differential Equations, 4th Edition (by Blanchard, Devaney, and Hall): https://amzn.to/35Wxabr. The concept of an m x n matrix with real number entries is defined. Matrix equality is defined. A linear transformation is defined by a matrix/vector product (matrix/vector multiplication). It can therefore be thought of as a matrix transformation. This is best defined by taking a linear combination of the columns of the matrix with the components of the vector as the weights (coefficients) in the linear combination. Low dimensional examples are considered and visualized. They are also described as onto (surjective) or not, and one-to-one (injective) or not. A linear transformation from R to R can be graphed in the usual way. A linear transformation from R to R^2 can be graphed as a parametric curve. A linear transformation from R^2 to R can be graphed as a surface in R^3 (three-dimensional space). And a linear transformation from R^2 to R^2 can be visualized as a mapping. #lineartransformations #matrixmultiplication (a.k.a. Differential Equations with Linear Algebra, Lecture 7B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 7B. #linearalgebra). Google drive link for Differential Equations and Linear Algebra course lecture documents: https://drive.google.com/drive/folder... Using Mathematica for ODEs (Ordinary Differential Equations) Playlist:    • Using Mathematica for Ordinary Differentia...   Visual Linear Algebra Online, Section 1.4 (at infinityisreallybig.com), Linear Transformations in Two Dimensions: https://infinityisreallybig.com/2019/... Visual Linear Algebra Online, Section 1.5 (at infinityisreallybig.com), Matrices and Linear Transformations in Low Dimensions: https://infinityisreallybig.com/2019/... Infinite Powers, How Calculus Reveals the Secrets of the Universe (by Steven Strogatz): https://amzn.to/2XXRCF6 Another Differential Equations lecture I made: Differential Equations: As Much As You Can Possibly Learn About in 50 Minutes, especially Population Models:    • Differential Equations Crash Course: As Mu...   Check out my blog: https://infinityisreallybig.com/ Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments. You can also get to know your professors personally. https://www.bethel.edu/ (0:00) Content goes with Section 1.5 of Visual Linear Algebra Online at Infinity is Really Big (0:53) Definition of an m x n real matrix (with real number entries) (2:08) Example 2 x 4 matrix with subscript notation for entries (3:08) Definition of matrix equality (3:44) Column vectors (m dimensional) are m x 1 matrices (4:44) Real numbers are 1 x 1 matrices (they can also be thought of as 1-dimensional vectors) (6:29) Linear transformations from R2 to R2 formula written with various notations (8:36) Operation-preserving property (“linearity property”) of linear transformations (11:22) Matrix representation of the formula for a linear transformation (13:46) The matrix product Ax as a linear combination of column vectors (the columns of the matrix A) with the components of x being the weights (coefficients) (16:50) This generalizes to higher dimensions (this is very important) (17:22) Definition of Ax, an m x n matrix A times an n-dimensional vector x as a linear combination of the columns of A with the components of x as the weights (19:56) Matching matrix dimensions and the dimension of the answer (20:34) Corresponding linear transformation from R^n to R^m (22:00) It is operation-preserving (22:25) Visualization of Linear Transformations T:Rn to Rm (22:51) Graphs of linear transformations from R to R (matrix of T is 1 x 1) (26:02) Image of T and whether it is an onto function (27:55) When is T a one-to-one function? (28:26) T:R to R^2 (29:31) Visualized as parametric curves (30:12) This will never be an onto function (30:35) It is “usually” a one-to-one function (30:54) Visualization as a motion in the plane (32:57) T:R^2 to R (34:50) The graph is a plane (surface) in three-dimensional space R^3 (35:58) This will never be a one-to-one function (36:52) It is “usually” an onto function (37:09) T:R^2 to R2 (37:29) Visualize as a mapping; it is “usually” both one-to-one and onto (38:48) In this case, one-to-one and onto are equivalent and related to the determinant of the matrix A (when the determinant is nonzero) AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.