Vector Space One Shot 🔥 | BSc/MSc | JAM, NET, GATE, CUET | Complete Vector Space for Semester Exams

Vector Space One Shot 🔥 Complete Vector Space for Semester Exams | BSc/MSc | JAM, NET, GATE, CUET TIMELAPS:- 00:00 Intro 00:19 Lecture Roadmap 01:30 Introduction to Vector Spaces 01:46 Applications of Vector Spaces 02:52 Definition (Start) 05:49 Basic Terms & Properties (Closure Property with Examples) 14:18 Definition (Continued) 15:40 Multiplication vs Scalar Multiplication 18:18 Definition (Continued) 23:30 Standard Notation of Vector Spaces 24:29 Vector Space Definition (Friedberg) – Term by Term Explanation 26:48 Question 1 (Theorem): In a Vector Space V over a Field F, (i) 0α = 0 for all α ∈ V (ii) c0 = 0 for all c ∈ F (iii) 1α = α for all α ∈ V (iv) cα = 0 implies either c = 0 or α = 0 59:12 Question 2: In any Vector Space V, show that (a + b)(x + y) = ax + ay + bx + by for any x, y ∈ V and any a, b ∈ F. 01:01:39 Geometric Interpretation of (a + b)(x + y) = ax + ay + bx + by 01:04:24 Question 3: Show that the set of all elements of the form a + b√2 + c∛3, a, b, c ∈ Q form a Vector Space over the field Q under usual addition and scalar multiplication of real numbers. 01:29:06 Question 4: Show that the set of all real convergent sequences is a Vector Space over the field of real numbers. 01:30:41 Understanding Sequences and Convergence with Examples 01:40:49 Solution of Question 4 02:01:26 Question 5: Show that the set of all real-valued continuous, differentiable or integrable functions defined on the interval [0,1] forms a Vector Space over the field of real numbers. 02:18:34 Question 6: Show that the set of matrices forms a Vector Space over the field of real numbers. 02:29:53 Question 7: Let R be the field of real numbers and let Pₙ be the set of all real polynomials of degree at most n. Then prove that Pₙ is a Vector Space over the field R. 02:33:00 Theorem: Uniqueness of Additive Identity and Additive Inverse in a Vector Space 02:34:48 Quick Recap of Key Concepts and Questions 02:36:57 Question 8: Let V be the set of all pairs (x, y) of real numbers, and let F be the field of real numbers. Define (x, y) + (x₁, y₁) = (x + x₁, 0), c(x, y) = (cx, 0). Is V with these operations, a Vector Space over the field of real numbers? 02:44:05 Question 9: Let V be the set of all pairs (x, y) of real numbers, and let F be the field of real numbers. Examine whether V is a Vector Space over the field of real numbers or not in each of the following cases. 02:53:26 Question 10: Let V be the set of all 3-tuples of the form (x, y, 1) with addition defined as (x₁, y₁, 1) + (x₂, y₂, 1) = (x₁ + x₂, y₁ + y₂, 1) and scalar multiplication defined by c(x₁, y₁, 1) = (cx₁, cy₁, 1), where c ∈ R. Which of the following is incorrect? 03:13:32 Question 11: Let S be a set of all matrices of the form [a 1; 1 b] with addition defined as [a 1;1 b] + [c 1;1 d] = [a+c 1;1 b+d] and scalar multiplication defined by k[a 1;1 b] = [ka 1;1 kb], where k,a,b,c,d ∈ R. Which of the following is incorrect? 03:18:02 Question 12: Let V denote the set of ordered pairs of real numbers. If (a₁,a₂) and (b₁,b₂) are elements of V and c ∈ R, define (a₁,a₂)+(b₁,b₂)=(a₁+b₁,a₂b₂) and c(a₁,a₂)=(ca₁,a₂). Is V a Vector Space over R with these operations? Justify your answer. 03:23:03 Question 13: Let V={(a₁,a₂)|a₁,a₂∈R}. Define addition of elements of V coordinatewise, and for c∈R and (a₁,a₂)∈V, define c(a₁,a₂)={(0,0), if c=0; (ca₁,a₂/c), if c≠0}. Is V a Vector Space over R with these operations? Justify your answer. 03:28:54 Question 14: Let V={(a₁,a₂)|a₁,a₂∈F} where F is a field. Define addition coordinatewise and c(a₁,a₂)=(a₁,0). Is V a Vector Space over F with these operations? Justify your answer. 03:30:24 Question 15: Let V be the set of ordered pairs (a,b) of real numbers. Show that V is not a Vector Space over R with addition and scalar multiplication defined by: (1) (a,b)+(c,d)=(a+d,b+c) (2) (a,b)+(c,d)=(a+c,b+d) (3) (a,b)+(c,d)=(0,0) (4) (a,b)+(c,d)=(ac,bd) 03:44:32 Question 16: A real-valued function f defined on the real line is called an even function if f(-t)=f(t). Prove that the set of even functions defined on the real line is a Vector Space. 03:58:37 Question 17: Which of the following is/are Vector Space(s)? (a) Polynomial Space (b) Functions with local maxima at x=1/2 (c) Matrix Space (d) Real-valued Functions 04:03:25 Question 18: Let U and W be Vector Spaces over a field K. Let V be the set of ordered pairs (u,w) where u∈U and w∈W. Show that V is a Vector Space over K with (u,w)+(u′,w′)=(u+u′,w+w′) and k(u,w)=(ku,kw). (External Direct Product) 04:07:37 Question 19: Which of the following is not a Vector Space? (a) C² over R (b) C² over Z (c) R² over C (d) R² over R 04:19:17 Question 20: Let V be the set of ordered pairs (a,b) of real numbers with (a,b)+(c,d)=(a+c,b+d) and k(a,b)=(ka,0). Show that V satisfies all the axioms of a Vector Space except 1u=u. 04:20:25 Summary of the Lecture 04:21:49 References 04:22:38 Thank You #VectorSpace #LinearAlgebra #Iitjam #Cuetpg #csirnet #Gate #SemesterExam #Bsc #Msc #oneshot #nitw #math #mathdil