Algebra davidad Relations Labels Composing Joining Inverting Commuting How I Think About Math Linearity Fields Part I: Linear Algebra “Linear” defined Vectors Matrices Tensors Subspaces David Dalrymple Image & Coimage [email protected] Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra March 6, 2014 CP decomposition Algebra Chapter 1: Relations davidad 1 Relations Relations Labels Labels Composing Joining Composing Inverting Commuting Joining Linearity Inverting Fields Commuting “Linear” defined Vectors 2 Linearity Matrices Tensors Fields Subspaces “Linear” defined Image & Coimage Kernel & Cokernel Vectors Decomposition Matrices Singular Value Decomposition Tensors Fundamental Theorem of Linear Algebra 3 Subspaces CP decomposition Image & Coimage Kernel & Cokernel 4 Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined 3 2 6 Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition 6 = 2 · 3 Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined 2.5 2 5 Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition 5 = 2 · 2.5 Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined 0 2 0 Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition 0 = 2 · 0 Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Kernel & Cokernel Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Only the Kernel & Cokernel directionality of the operator “2·” is significant. Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Only the Kernel & Cokernel directionality of the operator “2·” is significant. Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Analogously, writing y(x) is just politics: “x gets to tell y what to do!” Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Only the Kernel & Cokernel directionality of the operator “2·” is significant. Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y(x)= 2 · x Analogously, writing y(x) is just politics: “x gets to tell y what to do!” It can be useful to sequence computations hierarchically, but in the Platonic ideal world of mathematics, Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Only the Kernel & Cokernel directionality of the operator “2·” is significant. Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y = 2 · x Analogously, writing y(x) is just politics: “x gets to tell y what to do!” It can be useful to sequence computations hierarchically, but in the Platonic ideal world of mathematics, all variables are equal. Algebra A simple relation davidad Relations Labels Composing Relations are a generalization of functions; they’re actually more like constraints. Joining Inverting Here’s an example: Commuting Linearity Fields · “Linear” defined x 2 y Vectors Matrices Tensors Subspaces Image & Coimage Really, the directional annotations on the arrows are just that: annotations. Only the Kernel & Cokernel directionality of the operator “2·” is significant. Decomposition Singular Value Decomposition This might be more familiar to you as the equation: Fundamental Theorem of Linear Algebra CP decomposition y = 2 · x Analogously, writing y(x) is just politics: “x gets to tell y what to do!” It can be useful to sequence computations hierarchically, but in the Platonic ideal world of mathematics, all variables are equal have equal standing. Algebra A simpler relation davidad Relations Labels Composing Joining Inverting Commuting x y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simpler relation davidad Relations Labels Composing Joining Inverting Commuting x y Linearity Fields “Linear” defined Vectors You might better know this relation as Matrices Tensors Subspaces y = x Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simpler relation davidad Relations Labels Composing Joining Inverting Commuting 3 3 Linearity Fields “Linear” defined Vectors You might better know this relation as Matrices Tensors Subspaces 3 = 3 Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simpler relation davidad Relations Labels Composing Joining Inverting Commuting 2 2 Linearity Fields “Linear” defined Vectors You might better know this relation as Matrices Tensors Subspaces 2 = 2 Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simpler relation davidad Relations Labels Composing Joining Inverting Commuting 0 0 Linearity Fields “Linear” defined Vectors You might better know this relation as Matrices Tensors Subspaces 0 = 0 Image & Coimage Kernel & Cokernel Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A simpler relation davidad Relations Labels Composing