Review Sheet for Final Exam
Math 311
Dec. 5, 1995

1. Be sure you understand the following definitions, algorithms, and notations:
   1. C[0,1], , , , .
   2. Vector space, subspace, row space, null space, column space, rank, domain, range, image.
   3. Norm of a vector, dot product, projection, orthonormal basis, Gram-Schmidt procedure.
   4. Linear independent set, linear dependent set, spanning set, basis, dimension.
   5. Coordinates, change of basis matrix.
   6. Linear function, rank, matrix representation of a linear function, eigenvalue, eigenvector.
   7. Least squares types of problems.
   8. Polar, cylindrical, and spherical coordinate systems.
   9. Multiple and iterated integrals, change of variables formula.
   10. Line integrals, surface integrals.
   11. Stokes theorem and the divergence theorem.
   12. Local extremum, Hessian matrix, Lagrange multipliers.
   13. Grad(f), curl(F), and div(F).
2. You need to be able to do the following:
   1. Determine if a function is or is not linear.
   2. Solve a system of linear equations.
   3. Construct bases for the following subspaces: null space, row space, column space, range of a linear transformation.
   4. Construct matrix representations of linear transformations. For example rotations about a given axis, reflections through a plane.
   5. Diagonalize a matrix. Find an orthonormal basis of eigenvectors of a symmetric matrix.
   6. Given the velocity and/or acceleration vector of some object determine the actual path of the object.
   7. Given the path of an object determine the unit tangent and normal vectors associated with that path.
   8. Locate zeros of vector valued functions.
   9. Locate extremums, both global and constrained, of real valued functions.
   10. Calculation of work and flux due to a force field.
3. Sample problems:
   1. Let . Find a basis for the null space of A. Is the vector (1,2,-4,5) in the null space. Does the equation have a solution. Find a basis for the row space of A, the column space of A. What is the rank of A?
   2. Let . What are the eigenvalues and eigenvectors of the matrix A. Compute: , , , .
   3. Find a basis for the subspace .
   4. Given the data points . If a linear transformation from into has these data points in its image, is it possible to determine the dimension of its nullspace, the dimension of its column space? Explain.
   5. Find the matrix representation with respect to the standard basis of of the linear transformation which first rotates 44 degrees about the axis parallel to the vector (1,1,3) (The direction of rotation is clockwise when the origin is viewed from the point with coordinates (1,1,3).) and then reflects through the plane perpendicular to the vector (-2,3,7). If L is the symbol which represents this mapping, what is L(-2,5,4)?
   6. Find the shortest distance from the point (2,3,1,5) to the subspace

      . The notation denotes the standard inner product in .

   7. Find the linear combination of the functions , which best fits the data .
   8. Let . Let .
      1. Find all local and absolute extremum of the function f.
      2. Find all extremum of f subject to the condition .
      3. Let be a curve in such that . Suppose that . Calculate and determine .
   9. Let . Find those values of such that .
   10. Let Let .
       1. .
       2. .
       3. .
   11. Let . This rectangle in u-v space will be referred to as R. Denote this transformation by T. That is, .
       1. Graph T(R) in the x-y plane.
       2. Calculate the area of T(R).
       3. Show that this is an orthogonal transformation. That is, first find the vectors and , which are the unit vectors pointing in the direction of increasing u and v respectively, and then show that these two vectors are perpendicular to each other at every point in T(R)..

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