Mathematics 2 Week 6 Graded Assignment Solution

📏 Mathematics 2 Week 6 Graded Assignment

Versions: v1_t12023, v2_t22024 — Both versions contain a comprehensive, highly technical set of problems focused on advanced Linear Transformation concepts.

This downloadable PDF bundle provides complete, step-by-step solutions for Week 6 of “Mathematics for Data Science-2.” This module, Linear Transformations, Kernel, and Image, transitions from matrix properties to the geometric and algebraic effects of linear mappings, requiring a strong command of matrix representation with respect to non-standard bases.

Question Breakdown (11 Core Problem Parts):

• Q1: Linear Transformation Condition Test — Analyze a piecewise-defined function T: ℝ² → ℝ to test for additivity and homogeneity. Identify where linearity fails.
• Q2: Matrix Representation with Non-Standard Bases — Given T: ℝ³ → ℝ³ with identity matrix representation in bases β and γ, derive T(x, y, z), compute standard matrix A, and find det(A).
• Q3: Geometric Mapping of Subsets — Match four linear transformations to their image sets and geometric plots. Use substitution and matrix form to identify transformed spaces.
• Q4: LT to Transpose & Rank — For S: M₂(ℝ) → M₂(ℝ) defined by transpose, determine matrix representation B, its order, rank, nullity, and surjectivity.
• Q5: LT Injectivity/Surjectivity & Isomorphism — Given T(v) = Av with invertible A, use det(A) ≠ 0 and Rank-Nullity Theorem to confirm linearity, one-to-one, and onto properties.
• Q6: Basis of Null Space — For a 4×4 matrix A, reduce to RREF, find nullity, and identify basis vectors for Null Space ℕ(A).
• Q7: Rank & Nullity from Given Values — Given matrices A (4×3) and B (3×4) with known nullities, apply Rank-Nullity Theorem to compute ranks and maximum possible rank of AB.
• Q8: Basis of Skew-Symmetric Matrices & Nullity — For T: M → ℝ where M is the space of n×n skew-symmetric matrices, find dimension of M and nullity of T using trace properties.
• Q9–Q11: Comprehension – Linear Production Function:
  • Q9: Use T(x, y, z) to determine rank, nullity, and whether T is one-to-one or onto.
  • Q10: Evaluate kernel and image scenarios (e.g., no wheat used).
  • Q11: Calculate output for a new combination of ingredients.

File Format & Compatibility:

• File Type: PDF (Portable Document Format)
• v1_t12023, v2_t22024: Each version approx. 1 MB (11–27 pages)
• Solutions feature detailed derivations of Linear Transformation formulas, conversions between non-standard and standard bases, and rigorous application of the Rank-Nullity Theorem
• Optimized for the IIT Madras BS Degree program (Foundation/Diploma Level)