Take Home Midterm Exam On Linear Algebra 2020 Due Date

Take Home Midterm Exam On Linear Algebra 2020 Due Datetime 5pm F

Consider the tasks involving matrix algebra, invertibility, determinants, eigenvalues, and other properties of matrices. The exam includes computations such as elementary row operations for finding inverses, determinants, matrix partitioning for inversion, linear independence, span and rank, eigenvalues and eigenvectors, and definiteness of matrices. Answer each question thoroughly with appropriate calculations, justifications, and conclusions based on linear algebra principles. Use concepts such as elementary row operations, determinant criteria, eigenvalues and eigenvectors, matrix diagonalization, positive definiteness, negative definiteness, and semi-definiteness. Show all your work clearly and reference relevant theorems or formulas to substantiate your answers.

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Linear algebra forms a fundamental part of modern mathematics and provides essential tools for understanding systems of equations, transformations, and matrix properties. This exam encompasses various core concepts, from elementary row operations to eigenvalue analysis, illustrating their interconnected roles in matrix theory and applications.

Question 1: Inversion via Elementary Row Operations

Given a matrix A, the task is to perform elementary row operations to find its inverse. Suppose A is a 3×3 matrix; the process involves augmenting A with the identity matrix and applying row operations to transform the original matrix into the identity, simultaneously transforming the identity into A's inverse. Given the general form, one should systematically apply pivoting, row scaling, and row addition/subtraction procedures, ensuring the operations maintain equivalence to obtain A^-1. This method is grounded in the Gauss-Jordan elimination process, which guarantees an inverse if A is invertible, evidenced by the existence of a unique row-echelon form leading to the identity matrix.

Question 2: Determinant, Invertibility, and Norms

Part (a): For matrix A, computing the determinant involves expanding along rows or columns or applying properties such as row operations' effect on the determinant. If det(A) ≠ 0, then A is invertible; otherwise, it is singular. Judging invertibility follows directly from the determinant criterion.

Part (b): For A with det(A) = 2, the norms |3A|, |3A^-1|, and |(3A)^-1| are computed using properties of matrix norms, noting that multiplying a matrix by a scalar scales the norm by that scalar (or the scalar's absolute value). Specifically, |3A| = 3|A|, |3A^-1|= 3|A^-1|, and |(3A)^-1| = (1/3)|A^-1|.

Question 3: Matrix Partitioning and Invertibility

Part (a): Partitioning to compute the inverse involves expressing the matrix in blocks and applying block matrix inversion formulas, which utilize submatrix inverses and Schur complements. This method simplifies inversion especially for large matrices with known block structure.

Part (b): For a vector a in Rⁿ with ||a||=1, and vectors v₁, v₂, v₃ in R³, row reduction tests whether these vectors are linearly independent by forming a matrix with these vectors as columns and reducing it to row-echelon form. A set is linearly independent if the matrix has full rank; otherwise, dependencies exist.

Further, V = span{v₁,v₂,v₃} and its dimension is the rank of the matrix formed by these vectors. The matrix A formed by these vectors as rows or columns has rank equal to the dimension of V.

Question 4: Eigenvalues and Eigenvectors of Symmetric Matrices

For symmetric matrix A, eigenvalues are real, and eigenvectors corresponding to distinct eigenvalues are orthogonal. To find eigenvalues, solve (A - λI) = 0; eigenvectors are found by solving (A - λI)v=0 for each eigenvalue λ. Once eigenvalues and eigenvectors are known, orthogonal matrices H composed of normalized eigenvectors and a diagonal matrix D with eigenvalues satisfy H^TAH = D, basis for spectral decomposition.

Question 5: Positive Definiteness and Definiteness Tests

Assessing whether a matrix is positive semi-definite involves checking if all eigenvalues are ≥ 0, which can be done via eigenvalue computation or principal minors. For a parameter a, determining the signs of the eigenvalues or leading principal minors allows classification into positive definite, semi-definite, negative definite, semi-definite, or indefinite matrices. The conditions on a are derived from the positivity or negativity of these minors or eigenvalues.

Conclusion

This exam integrates a variety of fundamental linear algebra concepts, requiring both theoretical knowledge and computational skills. Mastery of elementary row operations, determinant calculations, eigenvalue problems, and matrix definiteness is essential for understanding matrix properties and their applications in advanced mathematics, physics, and engineering fields.

References

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