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 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 word about labels davidad
Relations Labels Composing Joining Inverting · Commuting x 2 y Linearity Fields “Linear” defined Vectors Matrices • Like the arguments of a subroutine, the labels of a relation are just a convenient Tensors “interface” for connecting it to a context or environment. Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra A word about labels davidad
Relations Labels Composing Joining Inverting · Commuting x 2 y Linearity Fields “Linear” defined Vectors Matrices • Like the arguments of a subroutine, the labels of a relation are just a convenient Tensors “interface” for connecting it to a context or environment. Subspaces Image & Coimage • Kernel & Cokernel If a label isn’t serving that purpose, we can remove it.
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Composing two relations davidad
Relations Labels Composing Joining Inverting Commuting 2· 1+ Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Composing two relations davidad
Relations Labels Composing Joining Inverting Commuting 2· 1+ Linearity Fields “Linear” defined Vectors Matrices Tensors This is way easier than composing functions. Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Composing two relations davidad
Relations Labels Composing Joining Inverting Commuting 2· 1+ Linearity Fields “Linear” defined Vectors Matrices Tensors This is way easier than composing functions. Subspaces Image & Coimage We just stick them together. Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Composing two relations davidad
Relations Labels Composing Joining Inverting Commuting 2· 1+ Linearity Fields “Linear” defined Vectors Matrices Tensors This is way easier than composing functions. Subspaces Image & Coimage We just stick them together. Kernel & Cokernel
Decomposition Sticking relations together like this will always give you a relation. Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors Subspaces x = y Image & Coimage Kernel & Cokernel Decomposition x = z Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors Subspaces x = y y = x Image & Coimage Kernel & Cokernel or Decomposition x = z y = z Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors Subspaces x = y y = x x = z Image & Coimage Kernel & Cokernel or or Decomposition x = z y = z z = y Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors Subspaces x = y y = x x = z Image & Coimage Kernel & Cokernel or or Decomposition x = z y = z z = y Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition They’re all the same! But with complex joins, this is easier to see in pictures. Algebra Joined Relations davidad
Relations Labels What does this mean? Composing Joining x y Inverting Commuting
Linearity Fields z “Linear” defined Vectors Matrices You could think of it as: Tensors Subspaces x = y y = x x = z Image & Coimage Kernel & Cokernel or or Decomposition x = z y = z z = y Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition They’re all the same! But with complex joins, this is easier to see in pictures. Relations with more than two “sides” (like this) are sometimes called
systems of equations.
But I find a single 3-sided relation more intuitive than a “system” of two equations. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces y 0.5· x Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces y 0.5· x Image & Coimage Kernel & Cokernel
Decomposition Singular Value Note: This is like the system of equations Decomposition Fundamental Theorem of Linear Algebra · CP decomposition y = 2 x
x = 0.5 · y Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces y 0.5· x Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces y 0.5· x Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, of Linear Algebra CP decomposition Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces x 0.5· y Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra CP decomposition Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces x 0.5· y Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces x 0.5· y Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting x 2· y Linearity Fields “Linear” defined Vectors Matrices Tensors Subspaces x 0.5· y Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x = 0 y Matrices Tensors Subspaces 3· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. Even this is valid if x happens to be 0. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. This diagram isn’t just true for some x... Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. This diagram isn’t just true for some x... it’s true for any x that is a “real” number, Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined R Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. This diagram isn’t just true for some x... it’s true for any x that is a “real” number, which we show like this. Algebra An example inverse davidad
Relations Let’s write “multiplication by 0.5 is the inverse of multiplication by 2.” Labels Composing Joining Inverting Commuting 2· Linearity Fields “Linear” defined R Vectors x y Matrices Tensors Subspaces 0.5· Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. This diagram isn’t just true for some x... it’s true for any x that is a “real” number, which we show like this. Algebra An example inverse davidad − Relations Let’s write “A 1 is the inverse of A over R.” Labels Composing Joining Inverting Commuting A Linearity Fields “Linear” defined R Vectors x y Matrices Tensors −1 Subspaces A Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition • Fundamental Theorem We can turn the bottom diagram around, like so. of Linear Algebra • CP decomposition Of course, x = x and y = y, so we can join those relations in. • But we can remove redundant labels, like so. Since we’re just going to transform y back into x anyway, we don’t even need a name for it. • The meaning is still imprecise. This diagram isn’t just true for some x... it’s true for any x that is a “real” number, which we show like this. Algebra Commutativity davidad
Relations Labels Composing Joining Inverting If we can reverse the order of two operators and get equal results, Commuting we say that they commute. Linearity Fields “Linear” defined Vectors Matrices Tensors 1+ 2+ Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Commutativity davidad
Relations Labels Composing Joining Inverting If we can reverse the order of two operators and get equal results, Commuting we say that they commute. Linearity Fields “Linear” defined Vectors Matrices Tensors 1+ 2+ Subspaces Image & Coimage Kernel & Cokernel = Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra 2+ 1+ CP decomposition Algebra Commutativity davidad
Relations Labels Composing Joining Inverting If we can reverse the order of two operators A and B and get equal results, Commuting we say that A and B commute. Linearity Fields “Linear” defined Vectors Matrices Tensors A B Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra Commutativity davidad
Relations Labels Composing Joining Inverting If we can reverse the order of two operators A and B and get equal results, Commuting we say that A and B commute. Linearity Fields “Linear” defined Vectors Matrices Tensors A B Subspaces Image & Coimage Kernel & Cokernel = Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra B A CP decomposition Algebra Commutativity davidad
Relations Labels Composing Joining Inverting If we can reverse the order of two operators A and B and get equal results, Commuting we say that A and B commute. We can express “A and B commute” like this: Linearity Fields “Linear” defined Vectors Matrices Tensors A B Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra B A CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra What is “Linear”? davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition Algebra
davidad
Relations Labels Composing Joining Inverting Commuting
Linearity Fields “Linear” defined Vectors Matrices Tensors
Subspaces Image & Coimage Kernel & Cokernel
Decomposition Singular Value Decomposition Fundamental Theorem of Linear Algebra CP decomposition