Linear Algebra: A Problem Based Approach

Clarifies the formal definition of matrix multiplication, deriving cij as the sum over k of aik bkj and showing the equivalence of notations through a 2-by-3 and a 3-by-3 example.

Identify square matrices, including 2x2 and 3x3 examples, and define diagonal, scalar, and identity matrices, as well as upper and lower triangular forms, and zero matrices.

Represent systems of linear equations with matrix notation, using m equations in n variables and augmented matrices; distinguish homogeneous from inhomogeneous cases and prepare for Gaussian elimination.

Master gaussian elimination to solve linear systems by row reducing to row echelon and reduced row echelon forms, and apply gauss-jordan to compute inverses from augmented matrices.

Use gaussian elimination to reveal an inconsistent system when a row reduces to 0 0 0 = -2. Note that rank(a) differs from rank(a'), explaining the no-solution outcome.

The theorem states that an n by n matrix is invertible iff its rank equals n; the matrix [[1, 2], [2, 4]] has rank 1, so it is not invertible.

Prove the uniqueness of the inverse matrix by showing that if AB=BA=I_n and AC=CA=I_n, then B=C, using associativity and the identity property.

Demonstrates that, given A = I − AB with B symmetric, A^T = A by applying the transpose and the uniqueness of the inverse.

Explore how av1 + bv2 forms a solution to Ax = b for an inhomogeneous system when a + b = 1, using matrix notation.

Explore how to compute a 3x3 determinant by expanding along any row or column, using the alternating plus minus pattern, and simplify with row operations and determinant properties.

Explore determinant properties: det AB = det A det B, det A^T = det A, det(lambda A) = lambda^n det A, and A is invertible iff det A ≠ 0.

Explore computing the determinant of a 4×4 matrix with a real parameter x, and practice generalizing to 2×2, 3×3, and N×N cases using row and column operations.

Demonstrate that for 2x2 real matrices, the trace of AB equals the trace of BA by comparing diagonal elements.

Apply Gaussian elimination to a three-variable system with a = -1, use row operations to reduce the matrix, and show the rank equals two, giving infinite solutions.

Study a real-parameter linear system with a and c to determine no, unique, or infinite solutions using Gaussian elimination, yielding row echelon form and pivot-based classification.

Define linear independence as condition that solution to lambda1 v1 + ... + lambdak vk = 0 in a vector space is the trivial solution, and that nontrivial solution implies linear dependence.

This solution for problem set 1 verifies vector space axioms through distribution and associativity proofs, then shows axiom ten fails with a counterexample, concluding V is not a vector space.

Prove linear independence of sin x and sin 2x by showing a sin x + b sin 2x = 0 for x implies a = 0 and b = 0.

Prove that the inverse of a bijective linear transformation is itself a linear transformation by showing additivity and homogeneity: t inverse(w1+w2) = t inverse(w1) + t inverse(w2) and t inverse(lambda w) = lambda t inverse(w).

Explore eigenvectors, eigenspaces, and geometric multiplicity: define eigenvectors via A v = λ v and eigenspace as null space of A - λ I, whose dimension is geometric multiplicity.

Analyze a 4x4 matrix with three eigenvalues, showing algebraic multiplicities equal geometric multiplicities and the characteristic polynomial factors, hence the matrix is diagonalizable.

Implement a python linear recurrence with x0=2, x1=3, x2=4; compute x_k from x_k=6 x_{k-1}-11 x_{k-2}+6 x_{k-3} and validate via matrix power A^k v0 using numpy.

Analyze eigenvalues and eigenvectors of C, show -6 has geometric multiplicity at least two and a zero eigenvalue, then conclude C is diagonalizable with diag(-6, -6, 0).