Learning Mathematics - Enjoy Pure Mathematics

 In the booklet Mathematical Reasoning  we discussed MCQ on Time and Distance, Simple and Compound Interest, Mixture and Alligation, Profit, Loss and Discount and Time and Work. This may be useful for any competitive examinations.  To download PDF of Mathematical Reasoning Click Here Dear Readers For Videos on Quantitative Aptitude visit the channel https://youtube.com/@m.velrajan-mdu?si=-sx5F0tCNEgJvxCy And for similar illustrations on Beauties of Numbers, Series, Numbers - Simplification, GCD and LCM, Linear Equations and Polynomials, Permutation, Combination and Inclusion and Exclusion, Geometry, Statistics, Data Interpretation, Age, Directions, Relationship, Coding and Arrangement visit my blogspot and get the PDF of my booklet “Quantitative and Reasoning Aptitude - First Step”. To visit the blogspot click the following link https://velrajanm.blogspot.com/2024/02/this-booklet-as-first-step-gives-some.html

Problem Solving - Linear Transformation   M.Velrajan   We continue our discussion on Problem Solving - Pure Mathematics with elementary exercises on linear transformations given in Topics in Algebra,  by I. N. Herstein, second edition. Let V be a finite dimensional vector space over a field F and  A(V) = Hom (V, V) be the algebra of all linear transformations of V into V. 1. S ∈ A(V) is regular if and only if whenever v 1 , . . . , v n ∈ V are linearly independent, then v 1 S, . . . , v n S are also linearly independent. Suppose S ∈ A(V) is regular.  Let v 1 , . . . , v n ∈ V be linearly independent.    To prove v 1 S, . . . , v n S are linearly independent we prove  𝛂 1 v 1 S + . . . + 𝛂 n v n S = 0 ⟹  𝛂 1 = . . . = 𝛂 n = 0. Then 𝛂 1 v 1 S + . . . + 𝛂 n v n S = 0           ⟹ (𝛂 1 v 1 + . . . + 𝛂 n v n ) S = 0 (since S is linear)         ...