4242: MATH Applied Linear Algebra Fall'08 Prototype for the final exam
Section 010
1. Compute the LU decomposition of a 33 matrix. Use it to compute the determinant of the matrix and to solve a linear system. [Midterm #1, Problem 2. Please have a look at the definitions of regular and nonsingular.] 2. A matrix is given as well as the result of transforming it to reduced upper echelon form. Use this form to find the rank of the matrix and bases for its range and corange. Find a basis for the kernel. Find dimensions of the four fundamental subspaces of the matrix. If we can find (by visual guess) one solution of the indeterminate system Ax = b and we have a basis for the kernel, computing the general solution of Ax = b is very simple. [Document Squeezing the matrix has the keys for much of this problem. Have a look at the part of Chapter 5 on compatible indeterminate systems. See also hints in the study guide.] 3. A 3 3 matrix K is given and x, y = x Ky is defined. Prove that this is an inner product (that is, prove that K is ... ). Find an orthogonal basis of a given subspace of R3 when the inner product is the above. Find the distance of a point to that subspace. [Midterm #2 Problem 5 is related to this problem. Review the GramSchmidt method, by the way. Using an orthonormal basis it is very simple to compute the closest point (a.k.a. orthogonal projection). When computing distances, don't forget to use the right notion of norm. The distance between two points is the norm of the difference of their vectors of coordinates] A 4. certain subspace W of P (2) is given using a characterization. I ask for a basis. Then I ask for a basis of its orthogonal complement W . [For bases of subspaces of polynomials, see Midterm #1, Problem 3 or Quiz #2, Problem 1. A vector is orthogonal to a subspace if and only if it is orthogonal to the elements of a basis of the subspace. This information should clarify how to find all the vectors of W .] 5. An affine function in R2 is given. I ask for a geometrical interpretation, the image of the unit square and the inverse. [Quiz #4, Problem 2. Have a look at the two sections of Chapter 7 dealing with concrete linear and affine transformations in the plane. They are very informative.] 6. A function L from P (2) to itself is given. I ask for you to show that it is linear. I give a basis and then I ask for the matrix representation and for the inverse of L. Finally I ask you to find p such that L[p] = q,...