MATH 4B Lecture Notes - Lecture 13: Coefficient Matrix, Linear Combination, Row And Column Vectors

• School: UCSB
• Department: Mathematics
• Course: MATH 4B
• Professor: Wei

Document Summary

Solutions are called the vector-valued functions because they output a vector in for each input. Solve for so that they satisfy the system of des. Solve coefficient matrix (given) inhomogeneous vector ( given) solution vector (unknown) coefficient matrix. If and are a pair of solutions, then any linear combination is also a solution. First order linear homogeneous system of equations in unknowns: There exists a set of solutions such that any solution is uniquely. These solutions are called a fundamental set of solutions. Form matrix by arranging the fundamental set of solutions in columnes: Alternative expression for general solution: fundamental matrix represented as a linear combination. Look for solutions in terms of exponential functions. General solution may also be written as , with is an matrix. Therefore, will be a solution if is an eigenvector of the coefficient matrix with eigenvalue. If an coefficient matrix has linearly independent eigenvectors with corresponding eigenvalues , then is a.