Mathematics 2 Week 3 Graded Assignment Solution

🔥 Mathematics 2 Week 3 Graded Assignment

Versions: v1_t12023, v2_t22024 — Both versions feature an identical problem set structure. The v1_t12023 provides 15 comprehensive problem parts.

This downloadable PDF bundle provides complete, step-by-step solutions for Week 3 of “Mathematics for Data Science-2.” The module, Introduction to Vector Space, is foundational, focusing on rigorous testing of vector space axioms, analyzing linear independence (LI) and linear dependence (LD), and applying these concepts to advanced models like Markov Chains in genetics.

Question Breakdown (15 Comprehensive Problem Parts):

• Q1: Vector Space Axioms Test — Evaluate four sets (V₁, V₂, V₃, V₄) with non-standard operations. Identify which axioms fail (e.g., Closure, Associativity, Zero Vector, Unity) to prove the set is not a vector space.
• Q2: Vector Space & LD/LI Properties — Prove distributive law, uniqueness of zero vector, and test LI/LD of vectors in ℝ³ and matrices in M₂×₂(ℝ) using determinant and zero matrix methods.
• Q3: Real-world Linear Combination — Model a nutritional profile vector in ℝ⁴ using a linear combination of ingredient vectors. Solve 4×3 or 4×4 SLEs using RREF to check for unique solutions.
• Q4: Linear Dependence Theorems — Apply five key theorems to a set of 5 vectors in ℝ³. Includes scalar multiple tests, determinant method, and homogeneous system analysis (Ax = 0).
• Q5: LD Parameter Test — Use determinant expression |A| = −2c + 5 to find the exact value of c that makes the vector set linearly dependent.
• Q6: LI Parameter Condition — Identify the value c must not equal for the set to remain linearly independent (det(A) ≠ 0).
• Q7–Q10: Comprehension – Genetics & Markov Chains:
  • Q7: Interpret transition matrix P to determine minimum guaranteed offspring traits (e.g., Round/Wrinkled Peas).
  • Q8 & Q9: Given X₀ = (1/3, 1/3, 1/3), compute X₁ = X₀P and X₂ = X₁P. Analyze whether sets {X₀, X₁} and {X₀, X₁, X₂} are linearly dependent or independent.
  • Q10: Given X₀ = (a, b, 1−a−b), find values of a and b that make X₂ = cX₀. Solve a system of non-linear equations to determine dependency.

File Format & Compatibility:

• File Type: PDF (Portable Document Format)
• v1_t12023: Approx. 5 MB (14 pages)
• v2_t22024: Approx. 9 MB (20 pages)
• Solutions feature rigorous, step-by-step proofs for vector space axioms and linear dependence. Includes full matrix reduction where necessary.
• Optimized for the IIT Madras BS Degree program (Foundation/Diploma Level)