Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4 Determine whether or not a matrix A is invertible. (Recall: we can test to see if A is invertible by looking at the RREF(A). If we get I for the RREF, A is invertible; otherwise A is not invertible. Use basic properties of inverses: (A ) = A, (AB) = B A, (A T ) = (A ) T Use inverses to solve an equation. Be able to compute the rank of a matrix. (Recall, the rank of A is the number of leading variables in the RREF(A). Section. Know what the null space (or kernel) of a matrix (or linear transformation) is. Be able to determine whether or not a vector x is in the null space of the matrix A. Know what the column space (or image) of a matrix (or linear transformation) is. Know what a linear combination of vectors is, and how to determine whether or not a given vector is a linear combination of a given set of vectors. Know what the span of vectors is, and how to determine whether or not a given vector is in the span of a given collection of vectors. Determine whether or not a vector x is in the column space of the matrix A. Know, and be able to use the fundamental ways to tell if an n n matrix A is invertible: (a) A can be row reduced to I. (b) The rank of A is n. (c) The system Ax = has only the trivial solution. (d) The nullspace of A is {}. (e) Every system Ax = b has a solution. (f) The column space of A is R n (g) The columns of A are a basis of R n. (h) There is a matrix B with BA = I. (i) There is a matrix C with AC = I. Section. Determine whether or not a given collection of vectors in R n is a subspace of R n. Know what it means for a collection of vectors to be linearly independent; linearly dependent; redundant and not redundant.
Be able to relate linear independence of the columns of A to the null space of A (i.e. the columns of A are linearly independent the null space of A is {}. Know what a basis for a subspace is. Determine if a set of vectors is a basis for a given subspace. Know that any basis for R n has n vectors; any n vectors of R n that span are automatically linearly independent; any n vectors of R n that are linearly independent automatically span. Find bases for the column space and the nullspace of a matrix. Section. Know what the dimension of a subspace is. Relate the column space, and null space of a matrix A to the corresponding spaces of the reduced row echelon matrix of A. Determine a basis for the column space and null space of a matrix A given the RREF(A). Determine the dimension of the nullspace (i.e. the nullity) and the dimension of the column space (i.e. the rank) of a matrix. Know that any two bases of a subspace have the same number of vectors; any n + or more vectors in an n-dimensional subspace are linearly dependent; and any n or fewer vectors cannot span an n-dimensional space. Be able to use the formulas: rank(a) + dim NUL(A) = number of columns of A, rank(a) = dim COL(A) = the number of pivot positions in A, and number of free variables in REF(A))= nullity(a). Be able to relate the solutions of a non-homogeneous system Ax = b to the solutions of the homogeneous system Ax =. Section.4 Be able to explain why if v, v,..., v n is a basis of S, then every v S can be uniquely written as v = c v + c v + + c n v n where c, c,..., c n are real numbers. Find the coordinates [x] B of a vector x with respect to a given basis B. Be able to find x if you know [x] B. Be able to find the matrix T B of a linear transformation with respect to basis B = {v, v,..., v n } in each of the ways we discussed. We will let S be the n by n matrix whose j column if v j, and let A be the matrix of T with respect to the standard basis. Through the translation diagram. In the standard basis e i goes to A(e i ); translating this into B-ese we multiply by S, This means S e i goes to S Ae i. So, if M is the matrix of T wrt to B we must have MS e i = S Ae i for all i. This happens when MS = S A, or equivalenty M = S AS. One column at a time. The j column of M is [T (e j )] B. As a matrix. M = S AS Be able to compute powers of A, and lim t A t knowing A = P MP and that M is a diagonal matrix. Know what it means for two matrices to be similar; and be able to show to matrices are or are not similar.
Section 5. Compute the length of a vector, the dot-product and angle between two vectors. Use the Cauchy-Schwarz inequality: If u and v are nonzero, then u v u v with equality if and only if u and v are parallel. Find the coordinates of a vector in terms of an orthogonal, or orthonormal basis. Find the projection of a vector onto a subspace with a given orthogonal or orthonormal basis. Section 5. Apply the Gram-Schmidt process to produce a basis of vectors to produce an orthogonal basis. Explain the idea behind the Gram-Schmidt process. Markov Chains Be able to determine the transition matrix M of a simple situation. Determine the distribution after or time-steps from the transition matrix and the initial distribution. Determine the long-term behavior by finding the equilibrium vector of M (the vector z with (M I)z = whose entries sum to ). Section 7. Know the definition of an eigenvector, and an eigenvalue. Given A and x be able to determine whether or not x is an eigenvector of A. Use information about the eigenvectors of one matrix to find eigenvectors and eigenvalues of a related matrix. Given A and scalar k be able to determine whether or not k is an eigenvalue of A, and if it is, find a corresponding eigenvector. Review Problems. A is an invertible matrix with A = (a) Use this to solve the equation below for x. Show your work. Ax = (b) Solve the following for Y: AY = A I.
. For which value(s) of k is the vector in the span of, k.. Below is the matrix A, and a matrix B obtained from A by row reduction. 4 A = 4 B = 6 6 4 4 (a) What is the rank of A? (b) Find a basis for the column space of A. (c) Show that is in the null space of (A). 8 6 (d) Find a basis for the null space of (A). (e) Verify (c), by writing as a linear combination of the basis found in (d). 4. The matrix A is a 5 by 5 matrix. List conditions that are equivalent to A being invertible. The conditions should be written in a precise manner. [ ] x 5. Let S be the collection of all vectors in R y with x y x. (a) List vectors that are in S (b) List vectors that are not in S. [ ] (c) Is in S? (d) Is S closed under scalar multiplication? Explain. (e) Find two vectors u and v with u and v in S, but u + v not in S. (f) Is S a subspace of R? Why or why not? 6. Let A by an n by n matrix. Show that the set of all n by n vectors with Ax = x forms a subspace of R n. 7. It is known that is a basis of R. (a) Find. B B =,,.
(b) Find y so that [y] B = 8. The matrix A is m by n, a basis of the null space of A is, and the rows of A are linearly independent.. (a) What s the nullity of A? (b) What s the rank of A? Explain your answer. (c) Determine m and n. Explain your answer. 9. Section., #8. Section., #4. Section., #9.. Section., #7.. Section., #5 4. Section., #8. 5. Section. #78. 6. Section.4 #5 7. Section.4 #4 8. Section.4 #,. 9. Section.4 #5, 55.. Section.4 #59.. Find the angle between the vectors u =, v = 4 5.. Section 5., line 6.. What should k be so that the vectors below are orthogonal? u = v = k k.
4. (a) Show that forms an orthogonal basis. (b) Find [x] B, where x =. B =,, 5. Section 5. #7. 6. Section 5. #9 7. A Christmas Party is held in a house whose floor plan is illustrated below. The caterer, from experience, knows that every 5 minutes the guests will are equally likely to move to one of the adjacent rooms or to stay in their current location. (a) Describe M so that x(t + ) = Mx(t) models this system. What do the components of M represent? (b) Find the equilibrium vector for M. (c) What, in every day language, do the entries of the equilibrium vector mean? (d) How should the caterer distribute the 6 trays of food that have been prepared (that is, how many trays should be put in each room)? Justify your answer. 8. Let A be an n by n matrix. Then an eigenvector of A is, and an eigenvalue of A is. 9. Let A = x =, and y = (a) Is x an eigenvector of A? (b) Is y an eigenvector of A? (c) Is an eigenvalue of A? If so, find a corresponding eigenvector.. Let where a is a real number. (a) For which value of a is an eigenvector of A? (b) For which a is an eigenvalue of A? A = a,
. (a) Show that if x is an eigenvector of the matrix A corresponding to the eigenvalue, then x is an eigenvector of A + I. What is the corresponding eigenvalue? (b) Show that if y is an eigenvector of the matrix A corresponding to the eigenvalue and A is invertible, then y is an eigenvector fo A. What is the corresponding eigenvalue? (c) Suppose that P is an invertible matrix, and R and S are matrices with P RP = S. Show that if z is an eigenvector of S with corresponding eigenvalue 7, then Pz is an eigenvector of R. What is the corresponding eigenvalue?