MA 527 first midterm review problems Hopefully final version as of October 2nd

Transcription

1 MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes and homework up to that date. Most of the problems on the exam will be closely based on ones from the list below (but the actual exam will be much shorter). For each problem, you must explain your reasoning. The last two pages are reference pages, which will also be provided on the actual exam. Note that these are not arranged in order of difficulty!. Consider the system of equations x + y + z = 0 x + y + 3z = a x + 3y + bz =. (a) For which values of a and b does it have no solution? (b) For which values of a and b does it have a unique solution? (c) For which values of a and b does it have infinitely many solutions?. Find a basis for the span of [5,, 4, 3,, [0, 0,, 0, 0, [0, 0, 0, 0,, [0,, 5, 6, 4, and [0,, 8, 6, Let A = [ (a) Find a basis for the column space of A. (b) Find a basis for the null space of A. (c) What is the rank of A? (d) What is the nullity of A?.

2 4. Suppose the matrix A = (a) Find the determinant of A. (b) Find the rank of A. (c) Find the nullity of A. a b c d e f g h i j k l m n p q (d) Find a basis for the column space of A. (e) Find a basis for the row space of A. (f) Find a basis for the null space of A. has row echelon form 5. Let A be a 4 6 matrix whose nullspace is spanned by (a) Find the rank of A. (b) Is A = 0? Explain your answer. (c) How many solutions are there to the equation Ax = your answer. 3 0 and ? Explain. 6. Let (a) Find det(a). A = [ 0 0.

3 (b) Is A invertible? If so, find A. 7. Let A = What is the determinant of A? Find matrices P, D, and P, such that P is orthogonal, D is diagonal, and [ 0 = P DP. 9. Find the general solution to the system x = 7x 4x, x = 8x 5x, and sketch a phase portrait. Label all equilibrium and straight line solutions. If you know that x (0) = and x (0) = 3, find x () and x (). 0. Suppose the system x (t) = Ax(t) has the following phase portrait. where the blue lines are given by x = x and x = x, and the matrix A has eigenvalues and. Find A.. Let α be a real number. For each of the systems below, and for each value of α decide which of the following terms apply to the description of 3

4 the origin: improper node, proper node, degenerate node, saddle point, center, spiral, stable, unstable, attractive. (a) x (t) = [ α x(t), (b) x (t) = [ α α 0 α x(t).. Sketch a phase portrait for the systems below near each equilibrium. (a) x = xy, y = xy + y x. (b) (c) x = xy, y = xy + y x. x = sin(πy) x, y = xy y + y. 3. Find the Laplace transform of (a) 0, 0 t <, f(t) = t, t <, 0, t <. (b) f(t) = te t. 4. Solve y + 3y + y = f(t), where {, 0 t <, f(t) = 0, t <, y(0) = 0 and y (0) = 0. Find y(00). What is lim t y(t)? 4

5 Reference pages The span of a set of vectors is the set of all possible linear combinations of those vectors. A set of vectors is linearly dependent if one of the vectors is a linear combination of the others; otherwise it is linearly independent. A basis of a vector space is a linearly independent set of vectors which span the space. The dimension of the space is the number of elements in a basis. The row space of a matrix is the span of the rows; column space is defined similarly. The rank of the matrix is the dimension of the row or column space. The nullspace of a matrix A is the space of solutions to Ax = 0. The column space is also the space of b such that Ax = b has a solution. The nullity of a matrix is the dimension of the nullspace. The rank and nullity of a matrix add up to the number of columns. If Ax = b has a solution, then the rank of A is the number of bound variables and the nullity is the number of free variables. If Ax = b has a solution x 0, then the general solution is x 0 + x h, where x h is the general solution to Ax = 0. Multiplying a row of a matrix by a constant multiplies the det. by the same constant. Switching two rows of a matrix multiplies the determinant by. (AB) T = B T A T and (AB) = B A. A square matrix is invertible if and only if it is nonsingular if and only if the determinant is nonzero. [ [ a b a b det c d = ad bc and c d [ a b c det d e f = aei + bfg + cdh ceg bdi afh. g h i [ = (ad bc) d b c a If A = P DP with D diagonal then D consists of the eigenvalues of A and the columns of P are the corresponding eigenvectors. If A is symmetric then P can be chosen orthogonal, which means P = P T and all the columns of P are unit length and mutually orthogonal. The general solution to y = Ay, where A is a diagonalizable x matrix and y a x vector is y = c e λ t v + c e λ t v, where v and v are independent eigenvectors of A and λ and λ are the corresponding eigenvalues. The origin is an equilibrium point of y = Ay. If the eigenvalues all have nonpositive real part, then it is stable, and if the real part is negative then it is attractive. If the 5.

6 eigenvalues are real and opposite signs it is a saddle, if they are real and same sign it is a node (improper if different, proper if the same and there are two eigenvectors). If the eigenvalues are complex it is a spiral, unless the real part is zero in which case it is a center. These properties all carry over to nonlinear systems, except that a center can become a spiral and a proper node can also become a spiral of the same stability/attractiveness. The general solution to y = Ay + g is y = y h + y p, where y h is the general solution to y = Ay and y p is a particular solution, which can be obtained by taking y p = Y u, where Y is a matrix whose columns are independent and each solve y = Ay, and where u = Y g. The Laplace transform is defined by L[f(t)(s) = F (s) = 0 e ts f(t)dt. f(t) F (s) t n n!s n e at f(t) F (s a) y sy (s) y(0) y s Y sy(0) y (0) cos bt s/(s + b ) sin bt b/(s + b ) u(t a)f(t a) e as F (s) 6