2431 Math Here are some of the topics discussed in this course: vectors in Rn , matrices addition, multiplication by a scalar product of matrices, and matrix vector; properties (linear maps = matrix maps) norm and dot product of vectors, angles and areas systems of linear equations: Ax = b row echelon form via Gauss elimination reduced row echelon form is unique nding solutions (a basis of solutions if b = 0) rank consistent and inconsistent systems if consistent: # of params = # unknowns rank A if b = 0, dimension of solution space = # unknowns rank A superposition (3.4) solutions for low-dimensional systems matrices, linearity, inverses linear maps ( = matrix maps) linear maps in R2 superposition (3.4, 4.7) computing inverses (via row reduction) solving Ax = b with A1 2 x 2 determinants and inverses; area(A(P )) = | det(A)| area(P ) systems of ODEs one ODE, initial value problems (x = f (x, t), x(t0 ) = x0 ) autonomous ODEs (x = g(x)); equilibria (g(x) = 0) and stability of hyperbolic equilibria (g (x0 ) = 0) x = x = x(t) = x0 et 2 x 2 systems (see much more in Chapter 6); sinks, sources, saddles graphic representations: time-series and phase-space portraits vector spaces denition subspaces (V vector space, W V ; W subspace W closed under addition and multiplication by scalars) e.g.: null(A) = {x Rn | Ax = 0}, {X (C 1 )n | X = CX} span of a family of vectors spanning sets linear independence dimension, bases Theorem 5.5.3, Corollary 5.6.7 1
dimension of a span and of a null-space: dim(span{w1 , . . . , wk }) = rank(M t ) = rank(M ), M = (w1 | . . . |wk ) dim(null(A)) + rank(A) = # of columns (i.e., # of variables) planar ODEs: X = CX, C an n n matrix ASIDE. general ODEs: X = F (X); F has continuous derivatives = uniqueness solutions of to IVP dimension of space of solutions = n (size of C), hence need n linearly independent solutions and use superposition solutions {X1 (t), . . . , Xn (t)} are linearly independent if so is the family of vectors {X1 (0), . . . , Xn (0)} direct method and matrix exponentiation direct method (6.2): three cases 1 , 2 real, two linearly independent eigenvectors 1 , 2 complex conjugate 1 = 2 real, one linearly independent eigenvector (use generalized eigenvector) matrix exponentiation: nd similarity to normal form (Thm. 6.5.5) determinants and eigenvalues (for n n matrices) properties: det(A) for A triangular, det(A) = det(At ), det(AB) = det(A) det(B) computing determinants: reduction to row echelon form or cofactor expansion A invertible det(A) = 0 characteristic polynomial pA () = det(A I) eigenvalues: roots of pA () = 0 eigenvectors: nonzero vectors with Av = v (hence is an eigenvalue) det(A) = 1 2 . . . n , trace(A) = 1 + + n det(A) = 0 no eigenvalue equal to zero A invertible = eigenvalues of A1 are the inverses of the ei...