devotion & mathematics

  For a video explanation, click here 👉             Theorem :      A singleton set with non-zero vector is Linearly Independent   Proof :           Suppose V ( F) is a vector space and S = {𝜶} is a subset of V where 𝜶 ≠ 0 .         Now we prove S is Linearly Independent.          For a 𝞊 F such that a𝜶 = 0                                      ⇒ a=0 or 𝜶 = 0  .          Since 𝜶 ≠ 0 , we have a = 0.            ∴ S is Linearly Independent.                                               ***Hence the proof ***      Explanation :           Linea...

  For a video explanation, click here 👉 https://youtu.be/NIQCW-Jg9yo                               Cayley-Hamilton Theorem :  Statement :     Every square matrix satisfies its characteristic equation Proof :        * Basis Extension Theorem   * V/W forms a vector space  *  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2  } forms a basis of V3 ( F )   

For a video explanation, click here 👉 https://youtu.be/YKNZT-lU7_w   Here in this page , i completely explained the original meaning of a continuous function at a point on the domain in detail.   CONTINUOUS FUNCTION :      Definition :   A real valued function f : X → R is said to be continuous at a point a in X if " for a given  𝞊 > 0 ∃ 𝝳 > 0 such that | f (x) - f(a) | < 𝞊 whenever | x-a | < 𝝳 .  now here is the explanation of the above definition.   Before going to the definition, we should learn two things.    1.  | x - y | means the distance between x and y.    2 .  | x-a | < r means x lies in circle with center a and radius r.    Note : Here circle means the neighbourhood.        For convinince or easily understand, here iam considering a neighbourhood in the real line  as a circle.                 ...

For a video explanation, click the link 👉 https://youtu.be/unnIkQ7DhuM      THEOREM :      Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of        all cosets forms a Vector Space under  the Coset addition and Scalar multiplication       are defined as        i.  For W + 𝛂 , W + 𝞫 𝞊 V,   ( W +  𝛂 ) + ( W +  𝞫 ) = W + ( 𝛂  +  𝞫 )       ii . For a 𝞊  F , W + 𝛂 𝞊 V,    a  ( W +  𝛂 ) =  ( W +  a𝛂 ).  PROOF :  * Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t.  x=(1,1,1),y = (-1,1,1),z = (1,0,-1)  *dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 )   *  If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷  *  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2...

For a video explanation, click the link 👉 https://youtu.be/1_g20kjooQs  QUOTIENT SET :        Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of        all cosets is called the Quotient Set in which the Coset addition and Scalar multiplication       are defined as        i.  For W + 𝛂 , W + 𝞫 𝞊 V,   ( W +  𝛂 ) + ( W +  𝞫 ) = W + ( 𝛂  +  𝞫 )       ii . For a 𝞊  F , W + 𝛂 𝞊 V,    a  ( W +  𝛂 ) =  ( W +  a𝛂 ). * Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t.  x=(1,1,1),y = (-1,1,1),z = (1,0,-1)  *dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 )   *  If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷  *  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2  } forms a basis ...

  📜 HISTORICAL BACKGROUND OF DIFFERENTIAL EQUATIONS 🌟 From Motion to Mathematical Equations Differential equations did not appear as an isolated branch of mathematics. They developed naturally from the study of motion, change, curves, mechanics, astronomy and physical phenomena . A differential equation relates an unknown function to one or more of its derivatives. Today, equations such as d y d x = f ( x , y ) \frac{dy}{dx}=f(x,y) are fundamental in mathematics, physics, engineering, economics and many other sciences. 🕰️ 1. The Beginning — Newton and Leibniz The history of differential equations is closely connected with the invention of calculus in the 17th century . 🔹 Isaac Newton (1642–1727) Newton developed his method of fluxions , thinking of quantities as things that flow or change with time . For example, if x x represents position, its rate of change can represent velocity: x ˙ = d x d t . \dot{x}=\frac{dx}{dt}. Newton applied these ideas to motion and mechanics . Hi...