MATH 211: Matrix Algebra I

LaTex files for assignments from UVic math 211, covering topics such as:

• Vector Spaces and Subspaces: Definitions, axioms, and properties such as closure under addition and scalar multiplication.
• Linearity and Linear Transformations: Proofs of linearity, computation of linear operators, and matrix representations.
• Matrix Operations: Solving systems of equations using Gaussian elimination and row reduction techniques.
• Eigenvalues and Eigenvectors: Calculating characteristic polynomials, eigenvalues, and eigenvectors.
• Special Matrices: Idempotent, symmetric matrices, and their transformations (e.g., reflections and rotations in 3D).
• Vector Calculations: Use of dot and cross products; understanding subspaces like orthogonal complements.
• Determinants and Polynomials: Computing determinants and using them for polynomial factorization.
• Applications: Practical scenarios like investment optimization modeled as linear systems.
• Advanced Transformations: Rotation and reflection matrices with geometric applications in linear algebra.
• Proof Techniques: Rigorous proofs involving properties of vector spaces, transformations, and matrix algebra.

Assignment 1 39/40

Question 1

• System of linear equations from financial scenarios.
• Solving systems of equations using substitution and elimination.

Question 2

• Analysis of under-determined systems: infinite solutions, unique solutions, and contradictions.
• Demonstration of consistent and inconsistent systems.

Question 3

• Polynomial interpolation and construction of a system of equations.
• Gaussian elimination and back-substitution for polynomial coefficients.

Question 4

• Parametric systems and their solutions.
• Conditions for no solutions, infinite solutions, and unique solutions using parameter analysis.

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Assignment 2 38/40

Question 1

• Subspace verification for vector spaces.
• Closure under addition and scalar multiplication.

Question 2

• Conditions for subspaces and failure cases.
• Counterexamples showing failure under scalar multiplication and addition.

Question 3

• Dot products and subspaces.
• Verifying closure, non-emptiness, and scalar multiplication for subspaces.

Question 4

• Dot and cross product applications in vector algebra.
• Numerical evaluations based on provided conditions.

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Assignment 3 38/40

Question 1

• Proof of linearity for operators (additive and scalar linearity).
• Constructing operator matrices for standard basis vectors.

Question 2

• Reflection, rotation, and transformation matrices.
• Combining transformations to find composite matrices.

Question 3

• Column space, null space, and rank-nullity theorem applications.
• Basis determination and span analysis.

Question 4

• LU factorization and solving systems using forward and backward substitution.
• Analyzing transformations and solving matrix equations.

Assignment 4 37/40

Question 1

• Vector Space Verification
• Vector Space axioms

Question 2

• Vector Space Properties
• Coordinates with respect to a basis for a vector space

Question 3

• Change of Basis Matrix
• Properties of change of basis matrix

Question 4

• Linear Transformations
• Verifying Linearity

Assignment 5 37/40

Question 1

• Computing Determinants
• Cofactor Expansion

Question 2

• Eigenvalues and Eigenvectors
• Proofs with Eigenvectors

Question 3

• Diagonalization & Diagonalizability