9: Variation of Parameters - Dissecting Differential Equations Mu Prime Math 2026-09-17 | 9: Variation of Parameters - Dissecting Differential Equations
Proof: Galois Group Maps Roots to Roots Mu Prime Math 2025-01-20 | An explanation for why automorphisms of a field extension map roots of a polynomial in the base field to other roots of that polynomial. This is a foundational fact in Galois theory that leads to basic properties of the Galois group of a field extension.Subscribe to see more new math videos!Music: C418 - Smooth Fall
Induction on Real Numbers Mu Prime Math 2024-10-25 | Video on supremum: youtu.be/o0TksrG5OsYPaper on real induction: arxiv.org/pdf/1208.0973Proof by induction is often taught as something that only works for integers, natural numbers, or whole numbers. However, there's an analogous concept to induction that applies to intervals on the real numbers, even though the real numbers are an uncountable set! This video explains and proves the result for the interval [0,1], which can be generalized to arbitrary closed intervals in the real numbers.Topology playlist: youtube.com/playlist?list=PLug5ZIRrShJEGnUPxM1KUVOHkW4kN6QU_0:00 The Theorem4:36 ProofSubscribe to see more new math videos!Music: C418 - Smooth Fall
The Derivative Equals The Square Mu Prime Math 2024-10-14 | We can write 1/(1-x) as the Taylor series 1+x+x^2+x^3+... But can we prove the simple identity d/dx (1/(1-x)) = 1/(1-x)^2 using only the infinite sum? The answer is yes, and it leads us to the more general idea of Cauchy products!Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao0:00 The Derivative2:14 The Square7:50 Generalization: Cauchy ProductsSubscribe to see more new math videos!Music: C418 - Smooth Fall
I Want to Play a Game: Proving [0,1] is Uncountable Mu Prime Math 2024-10-11 | The interval [0,1] is uncountable, and there are many ways to prove that. Cantor's diagonal argument is a classic example. But there's another way: proof by playing a game. This video is an explanation of a proof by contradiction after assuming that the infinite set [0,1] is countable and playing a game on that interval. All we need is basic calculus for the limits of sequences! Original based on Grossman & Turett: arxiv.org/pdf/math/0606253Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: C418 - Smooth Fall
Matrix Proof: det(exp A) = exp(Tr A) Mu Prime Math 2024-10-07 | Upper triangular matrices video: youtu.be/p0_3BDvf9JADid you know about this very nice equation with the matrix exponential? In this video we prove the result by reducing to the upper triangular case and connect it to basic identities of the exponential function. Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcs0:00 Reducing to Upper Triangular8:53 Main ProofSubscribe to see more new math videos!Music: C418 - Smooth Fall
Multiplying Upper Triangular Matrices Mu Prime Math 2024-10-04 | A proof that the product of upper triangular matrices is upper triangular. We also prove that the diagonal entries of the matrix product are simply the products of the diagonal entries of the factor matrices.Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcs0:00 Upper Triangular Definition1:25 Product is Upper Triangular6:42 Diagonal EntriesSubscribe to see more new math videos!Music: C418 - Smooth Fall
Does the Gaussian integral trick work for other functions? Mu Prime Math 2023-04-14 | Proof for Riemann-integrable functions:Robert J. MacG. Dawson. “On a “Singular” Integration Technique of Poisson”. American Mathematical Monthly, 2005. cs.smu.ca/~dawson/Poisson.pdfWe can compute the integral of e^(-x^2) using a very cool trick that lets us switch to polar coordinates and use the Jacobian for a u-sub. But does this technique work for any other function? In this video we turn that problem into a differential equation and find all of the solutions!Mariano Suarez-Alvarez's proof: math.stackexchange.com/a/4517530/7135470:00 The Functional Equation4:23 The Differential Equation7:14 Solving12:34 Additional NotesCalculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: C418 - Pr Department
Proof: Uniqueness of the Tensor Product Mu Prime Math 2023-04-07 | Universal property introduction: youtu.be/vZzZhdLC_YQThis video proves the uniqueness of the tensor product of vector spaces (or modules over a commutative ring). This uses the universal property of the tensor product to prove the existence of an isomorphism (linear bijection) between any two "tensor products".Tensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk0:00 The Universal Property3:53 Relating Two Tensor Products8:10 Proving IsomorphismSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Tensor Product Basis With the Universal Property Mu Prime Math 2023-03-31 | Tensor product universal property explanation: youtu.be/vZzZhdLC_YQGenerating set proof: youtu.be/KnSZBjnd_74?t=1437 timestamp 23:57If we have a basis for each of two vector spaces (or modules over a commutative ring) V and W, then we can use that to form a basis for the tensor product V⊗W. The proof uses the universal property of the tensor product, which connects bilinear maps on the Cartesian product to linear maps on the tensor product. This video explains how we can use the universal property to prove the tensor product basis. The goal is to improve our understanding of the applications of the universal property.Tensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk0:00 Introduction1:43 How to Prove Linear Independence6:18 Using the Universal Property12:11 ConclusionSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Complete Derivation: Universal Property of the Tensor Product Mu Prime Math 2023-03-23 | Previous tensor product video: youtu.be/KnSZBjnd_74The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universal property can be confusing if it is presented without background. This video is an explanation of the universal property that proves it for a concrete instantiation of the tensor product of vector spaces (or modules over a commutative ring).Tensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk0:00 Introduction3:04 Constructing the Tensor Product7:54 Bilinear Maps10:39 Maps on the Tensor Product16:17 Defining g26:24 Linearity and Uniqueness29:44 Universal Property30:50 ExampleSubscribe to see more new math videos!Music: C418 - Pr Department
How to Define Homomorphisms on Quotient Groups Mu Prime Math 2023-01-30 | One way to define a function on a quotient group is to define it in terms of the coset representatives. However, this approach runs into problems of well-definedness. This video explains how we can address that problem.0:00 Problem Introduction5:42 Solution10:34 Non-example: Integers mod 511:13 Example: Alternating GroupGroup Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6Subscribe to see more new math videos!Music: C418 - Pr Department
Why can we change lim n→∞ to lim x→∞? Mu Prime Math 2023-01-27 | When computing the limit of a sequence, it's often useful to consider a limit of real numbers so that we can do things like take derivatives. But why is this allowed in the first place? Why can we change a limit of integers to a limit of real numbers? This video gives an explanation.Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: OcularNebula - The Lopez
A Natural Proof of the First Isomorphism Theorem (Group Theory) Mu Prime Math 2023-01-09 | The first isomorphism theorem is one of the most important theorems in group theory, but the standard proof may seem artificial, like every step of the proof is set up knowing that we're trying to create an isomorphism. In this video, we show an alternate proof with no such tricks using the preimage map of a group homomorphism.Group theory playlist: youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6Subscribe to see more new math videos!0:00 Setup6:18 Homomorphism8:50 Injective10:35 SurjectiveMusic: C418 - Pr Department
Integral Formula for Natural Log (without knowing the derivative) Mu Prime Math 2023-01-03 | This video proves that the natural log equals the integral from 1 to x of 1/t dt under the assumption that ln(x) is the inverse function to the exponential e^x. We can do this without already knowing the derivative of the natural log!More details on why the integral is the inverse of e^x:We proved in the video that any right inverse to e^x must equal that integral. However, we didn't prove that e^x has a right inverse in the first place.We know that e^x : R → R+ is a strictly increasing function whose output can be made arbitrarily large or arbitrarily small. Therefore e^x is a bijection R → R+. Every bijective function has a two-sided inverse (see [1] below). Therefore e^x has a two-sided inverse, which in particular is a right inverse. I proved in video [2] that injective functions have at most one right inverse. Therefore the right inverse to e^x is unique if it exists. But we already know that there exists one right inverse that is also a two-sided inverse. We conclude that there exists exactly one right inverse to e^x and that this right inverse is also a two-sided inverse. Hence the integral in the video is a two-sided inverse to e^x.[1] youtu.be/E-njuKKrOwg[2]: youtu.be/i5ZJuJr8DwECalculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Supremum, Infimum: Definition and Explanation Mu Prime Math 2022-04-23 | Not all sets of real numbers have a maximum and a minimum. In this video, we introduce the supremum and infimum as a generalization of max and min. Sup and inf appear everywhere in analysis.Subscribe to see more new math videos!Music: OcularNebula - The Lopez
Topology Definitions: Connected and Path-Connected Mu Prime Math 2022-04-16 | Connectedness is a key idea in topology and metric spaces that describes whether a topological space can be separated into two components. This video explains the open set definition of connectedness and describes the topologist's sine curve as a non-example that motivates the definition of path-connectedness.Topology playlist: youtube.com/playlist?list=PLug5ZIRrShJEGnUPxM1KUVOHkW4kN6QU_Timestamps:0:00 connectedness9:59 path-connectednessSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Does 1+2+3+... 1/12? Mu Prime Math 2022-04-09 | The answer to the question in the title is that it depends on how you define the infinite sum function. Until the infinite sum function is specified, the question is not well-defined; it's the equivalent of asking "what is f(5)?" without defining the function f.The sum of all natural numbers, and the sums of divergent series more generally, are often debated because it seems incoherent to say that a divergent sum equals a finite number. This video explains the machinery used to obtain finite outputs from divergent infinite series.Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: C418 - Pr Department
Proof: Orthogonal Matrices Satisfy A^TA=I Mu Prime Math 2022-04-02 | One way to characterize orthogonal matrices is to say that a matrix orthogonal if and only if A transpose times A is the identity matrix. In this video, we prove this result using basic matrix calculations and the definition of orthonormal vectors.Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcsSubscribe to see more new math videos!Music: C418 - Pr Department
My favorite proof of the n choose k formula! Mu Prime Math 2022-02-19 | The binomial coefficient shows up in a lot of places, so the formula for n choose k is very important. In this video we give a cool combinatorial explanation of that formula!Challenge Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGkzGsXMYQt8bi5ImYtiEMMSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Epsilon-Delta proofs: Cant we make the limit equal anything? Mu Prime Math 2022-02-12 | Epsilon-Delta definition explanation: youtu.be/JbbRaiXI6ywProofs that use the delta-epsilon definition of the limit can be confusing because it seems like we can prove that the limit is anything we want if we pick the right value of delta. In this video we prove that limits are unique and go over some examples of disproving limits!Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao0:00 Background3:46 Example 19:46 Example 215:45 Backward proof?17:14 Proof of uniquenessSubscribe to see more new math videos!Music: C418 - Pr Department
Why cant we group the terms in 1-1+1-1+... ? Mu Prime Math 2022-02-05 | We can group the terms in the divergent infinite sum 1-1+1-1+... to get a value of zero or a value of one, even though the series doesn't converge. This video gives an explanation of why grouping terms in Grandi's series doesn't give the correct answer.Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao0:00 Why the sum diverges7:33 Why grouping fails12:57 When grouping is validSubscribe to see more new math videos!Music: C418 - Pr Department
When can we switch the limit and function? Mu Prime Math 2022-01-29 | Epsilon-delta definition explanation: youtu.be/JbbRaiXI6ywBringing functions outside of a limit is very useful for calculating limits. But when does this rule work? This video gives a proof and explanation of why we can switch limit and function if the outside function is continuous and the inner limit exists.Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Topology Definitions: Closure, Boundary, Interior Mu Prime Math 2022-01-22 | An explanation of how to define closure, boundary, and interior in topology using open and closed sets instead of a metric. Also explains adherence points. Intended as an introduction to basic concepts in topology.Topology playlist: youtube.com/playlist?list=PLug5ZIRrShJEGnUPxM1KUVOHkW4kN6QU_0:00 Intro1:24 Closure7:46 Boundary11:05 InteriorSubscribe to see more new math videos!Music: C418 - Pr Department
Why can we do this to find inverse functions? Mu Prime Math 2022-01-15 | One way to find the two-sided inverse of a function is to solve the equation f(x) = y for the variable x. But why does this always give us an inverse function? In this video, we give a proof and explanation of where this method comes from.Functions playlist: youtube.com/playlist?list=PLug5ZIRrShJG35bSmijiKFxerMar8jMCiSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Bijective Functions Have a Two-Sided Inverse Mu Prime Math 2022-01-08 | A proof that bijections have two-sided inverses by considering the preimages of injective and surjective functions.Functions playlist: youtube.com/playlist?list=PLug5ZIRrShJG35bSmijiKFxerMar8jMCiSubscribe to see more new math videos!Music: C418 - Pr Department
Proof: Two-Sided Inverse Functions are Unique Mu Prime Math 2022-01-01 | An explanation of why a function has at most one two-sided inverse. To prove this result, we prove that an injective function has at most one right inverse.Functions playlist: youtube.com/playlist?list=PLug5ZIRrShJG35bSmijiKFxerMar8jMCiSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Sum of natural numbers equals n+1 choose 2! Mu Prime Math 2021-09-24 | There is a formula for the sum of the first n natural numbers, which is n(n+1)/2. But that's the same thing as n+1 choose 2! Why are those the same? In this video we give a proof of that connection!Challenge Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGkzGsXMYQt8bi5ImYtiEMMSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Representation Theory: Irreducible Characters Are an Orthonormal Basis Mu Prime Math 2021-09-20 | A key result in complex representation theory is the fact that, under a specific Hermitian inner product, the characters of the irreducible representations of a finite group G are an orthonormal basis for the vector space of complex-valued class functions on G. This video is an explanation and proof of why the irreducible characters are an orthonormal basis.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 Orthonormal10:19 Basis21:15 SummarySubscribe to see more new math videos!Music: C418 - Pr Department
Matrix Invertibility With F[x]-Modules Mu Prime Math 2021-09-17 | Characteristic polynomial explanation: youtu.be/jCt6mR3QtPkRing & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7In this video, we use F[x]-modules to prove that a matrix is invertible if and only if its determinant is nonzero. Another example of how F[x]-modules can be applied to a ton of things in linear algebra!Subscribe to see more new math videos!Music: OcularNebula - The Lopez
Difference Between Normalizer, Centralizer, and Stabilizer Mu Prime Math 2021-09-13 | An easy way to remember what is the normalizer and centralizer of a subgroup, and what is the stabilizer of an element under a group action. For people learning abstract algebra!Group Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ6a6Subscribe to see more new math videos!
A Concrete Introduction to Tensor Products Mu Prime Math 2021-09-10 | The tensor product of vector spaces (or modules over a ring) can be difficult to understand at first because it's not obvious how calculations can be done with the elements of a tensor product. In this video we give an explanation of an explicit construction of the tensor product and work through several example computations, such as finding a generating set.Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcsTensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk0:00 Construction14:01 Examples23:33 Basis for Tensor Product28:44 ExamplesSubscribe to see more new math videos!Music: C418 - Pr Department
Eulers Identity With a Limit! Mu Prime Math 2021-09-06 | e^x limit for real numbers: youtu.be/HM-kwHR4VO4Euler's formula for the complex exponential can be proved in many different ways, such as with Taylor series. Here we find Euler's identity using a limit equation for e^x and the polar form of complex numbers!Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITaoSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Why Are Odd Numbers Not Even? Mu Prime Math 2021-09-03 | There are even numbers and odd numbers. But for some reason, there aren't any numbers that are even and odd at the same time. Why is that? In this video we give a proof that odd numbers are not even!Challenge Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGkzGsXMYQt8bi5ImYtiEMMSubscribe to see more new math videos!Music: C418 - Pr Department
The Cayley-Hamilton Theorem is Easy with F[x]-Modules Mu Prime Math 2021-04-27 | Invariant factors proof: youtu.be/gWIRI43h0icCharacteristic polynomial explanation: youtu.be/jCt6mR3QtPkIntro to F[x]-modules: youtu.be/H44q_Urmts0The Cayley-Hamilton theorem says that every matrix is a root of its own characteristic polynomial, det(xI-A). With all of our knowledge of F[x]-modules, the proof is simple! This video is an explanation of how we reach this important result in linear algebra.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 Proof6:32 ExampleSubscribe to see more new math videos!Music: OcularNebula - The Lopez
How the Matrix Characteristic Polynomial is Connected to F[x]-Modules Mu Prime Math 2021-04-25 | Rational canonical form: youtu.be/q5uj4o0O5R0Similar matrices isomorphism: youtu.be/-ligAAxFM8YIntro to F[x]-modules: youtu.be/H44q_Urmts0The characteristic polynomial described by the determinant det(xI-A) is very useful when studying matrices. It turns out that it's related to modules as well! Here we show how the characteristic polynomial is related to the module form of an arbitrary matrix.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 Computing the determinant10:07 Similar matrices have same polynomial12:40 Proof for a general matrixSubscribe to see more new math videos!Music: C418 - Pr Department
F[x]-Module Derivation of Rational and Jordan Canonical Forms Mu Prime Math 2021-04-23 | Similar matrices isomorphism proof: youtu.be/-ligAAxFM8YEvery module is a direct sum of cyclic modules: youtu.be/gWIRI43h0icIntro to F[x]-modules: youtu.be/H44q_Urmts0The rational canonical form and Jordan normal form of a matrix are very important tools in linear algebra, but ring theory and module theory give us a very effective way to prove their existence! Here we show that every matrix is similar to a matrix in rational and Jordan canonical form.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 Rational canonical form14:17 Every matrix is similar to RCF18:33 Algebraically closed fields20:10 Jordan canonical formSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Proof & Intuition: Similar Matrices are Isomorphic F[x]-Modules Mu Prime Math 2021-04-21 | Intro to F[x]-modules: youtu.be/H44q_Urmts0Modules over a polynomial ring are powerful tools to study linear algebra. This video is an explanation of why similar matrices are related to isomorphisms of F[x]-modules!Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7Subscribe to see more new math videos!Music: OcularNebula - The Lopez
Introduction and Derivation of F[x]-Modules Mu Prime Math 2021-04-19 | Modules over a polynomial ring have very important applications to linear algebra. Here we prove some basic properties of F[x]-modules and show how they are related to vector spaces over a field.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7Subscribe to see more new math videos!Music: C418 - Pr Department
What do Matrices Represent? - Learning Linear Algebra Mu Prime Math 2021-04-17 | This video is about why we use matrices and how every matrix is related to a function that takes vectors as inputs. Understanding what a matrix represents is important in order to learn about the more advanced ideas in linear algebra!Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcsSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Proof: Structure Theorem for Finitely Generated Torsion Modules Over a PID Mu Prime Math 2021-04-15 | This video has chapters to make the proof easier to follow.Splitting explanation: youtu.be/ZINtBNje_08In this video we give a proof of the classification theorem using two smaller proofs by induction. We show both the elementary divisor form and the invariant factor form of a module. This theorem tells us a lot about modules over a principal ideal domain!Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 First lemma2:15 First lemma proof8:51 Second lemma9:24 Second lemma proof11:20 Proof for p^r28:30 Final proof36:35 End of proof (additional explanation)Subscribe to see more new math videos!Music: C418 - Pr Department
Splitting Homomorphism of R-Modules Mu Prime Math 2021-04-13 | A splitting, or section, is a homomorphism from the quotient module to the original module that gives a representative for each coset. If we have a splitting, we can prove that the module is isomorphic to a direct sum! This video is an explanation of how the splitting leads to an isomorphism.Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7Subscribe to see more new math videos!Music: C418 - Pr Department
Proof: Prime Ideals are Maximal in a PID Mu Prime Math 2021-04-11 | In a principal ideal domain, if an ideal is a prime ideal, that implies it is a maximal ideal, as long as it is not just the zero ideal. Here we give a straightforward explanation of this theorem from ring theory!Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7Subscribe to see more new math videos!Music: C418 - Pr Department
Proof: Prime Ideal iff R/P is Integral Domain; Maximal iff R/M is Field Mu Prime Math 2021-04-09 | A very useful theorem in ring theory is the theorem that an ideal P is prime if and only if the quotient R/P is an integral domain (ID). Similarly, an ideal M is maximal if and only if R/M is a field. In this video, we prove both of these statements!Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq70:00 Prime ideal3:06 Maximal ideal9:16 Maximal implies primeSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Why Simultaneity MUST Be Relative Mu Prime Math 2021-04-07 | Derivation of Lorentz transformation: youtu.be/6f_yxbtM2TIRelativity of simultaneity can be very confusing the first time you see it because it doesn't match everyday experience. However, it turns out that we need simultaneity to be relative for physics to work! Here we use linear algebra to explain why events that are simultaneous in one reference frame are not simultaneous in another reference frame.Special Relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0qSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Length Contraction, Time Dilation, & Relativity of Simultaneity Using Lorentz Transformation Mu Prime Math 2021-04-06 | Lorentz transformation derivation: youtu.be/6f_yxbtM2TISpecial relativity gives us a lot of unintuitive results. We can understand why these happen using the Lorentz transformation, so we can see how coordinates in one inertial reference frame appear to an observer in a different reference frame!Special Relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0q0:00 Length contraction5:44 Relativity of simultaneity10:08 Time dilationSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Why ct² - x² is Invariant under Lorentz Transformation Mu Prime Math 2021-04-04 | Derivation of the Lorentz transformation matrix: youtu.be/6f_yxbtM2TIThe spacetime interval s^2 = ct^2 - x^2 is important in special relativity because it stays the same in all inertial reference frames. Here we use some linear algebra to give a proof of exactly why this interval is invariant!Special Relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0qSubscribe to see more new math videos!Music: C418 - Pr Department
Linear Algebra Derivation of Lorentz Transformation Mu Prime Math 2021-04-02 | Why the Lorentz transformation is linear: youtu.be/5uFfrmhKeQUWhy the eigenvalues need to be positive: We can write the vector (β,1) as a linear combination of the two eigenvectors. If one of the eigenvalues is negative, then (β,1) will get flipped along one of the axes of the eigenvectors. Because |β| is less than 1 we know that (β,1) starts out in the top region, above y=x and y=-x. Once it gets flipped it won't be in the top region anymore. In that case, it can't possibly equal (0,1) because (0,1) is in the top region. This contradicts our assumption. Therefore the eigenvalues can't be negative because that leads to a contradiction. The eigenvalues also can't be zero because then there would be a nonzero null space, so the map would not be injective (multiple values map to zero), so it would not have an inverse. As a result, we know that the eigenvalues must be positive.The Lorentz transformation is often derived using thought experiments about shooting light rays and things like that. But we can prove the matrix form of the Lorentz transformation in a more abstract way using linear algebra! This gives us a way to describe changes of coordinate system when we move between inertial reference frames.Special relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0q0:00 Starting assumptions6:20 DerivationSubscribe to see more new math videos!Music: C418 - Pr Department
Intuition: Why the Lorentz Transformation is Linear Mu Prime Math 2021-04-01 | Main Lorentz Transformation video: youtu.be/6f_yxbtM2TIDr Peyam video on f(x+y)=f(x)+f(y): youtu.be/WnglFnfjjFsSpecial relativity intro: youtu.be/upfIW5Ci0mQOne of the most important starting points for deriving the Lorentz transformation is the fact that it is a linear transformation. Why do we get to assume that? Here we explain the intuition behind why we expect the Lorentz transformation to be linear, using the coordinate systems of special relativity and how rulers and clocks change between inertial reference frames.Special Relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0q0:00 Introduction4:48 T(u+v) = T(u)+T(v)15:13 T(cv) = cT(v)Subscribe to see more new math videos!Music: OcularNebula - The Lopez
Special Relativity Intro: Reference Frames & Spacetime Diagrams Mu Prime Math 2021-03-30 | Special relativity gives us a powerful new way to think about physics. In order to apply special relativity, we first have to understand coordinate systems! Here we talk about defining coordinate systems using rulers and clocks, as well as using spacetime diagrams to draw world lines.Special Relativity playlist: youtube.com/playlist?list=PLug5ZIRrShJHWM8_CPL5Nh5mPlEUNor0q0:00 Reference frames5:19 Spacetime diagramsSubscribe to see more new math videos!Music: OcularNebula - The Lopez
Derivative of Area is Perimeter? Mu Prime Math 2021-03-29 | Epic Math Time's video: youtu.be/0vYWsOBBXxwWhen is the derivative of area equal to the perimeter? In this video we solve this calculus/geometry problem for any shape. Then we look at regular polygons and get some very interesting results!Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao0:00 General case6:21 Examples9:26 Regular polygon caseSubscribe to see more new math videos!Music: C418 - Pr Department
