The following list indicates the things you should be able to do for Exam 2 on Thursday, March 9th. In most cases you can find examples in the text or the class notes. The list is not meant to be entirely exhaustive - anything we've done in class or on homework assignments or quizzes is "fair game."

• Basic Tests and Computations:
  • Determine whether a vector is in the span of a set of vectors.   4.1: 2,   Assignment 9 Exercise 2, solutions here
  • Determine whether a vector is in the column space of a matrix.   4.4: 1,   Assignment 9 Exercise 6, solutions here
  • Determine whether a vector is in the null space of a matrix.   4.4: 2,   Assignment 9 Exercise 5, solutions here
  • Determine whether a set of vectors are linearly independent. If they are, give one of them as a linear combination of the others.   4.6: 1, 2, 3,   Assignment 9 Exercises 3, 4, solutions here
  • Give a basis for the column space of a matrix.   4.8: 3, 4,   Assignment 9 Exercise 7(b), solutions here
  • Give a basis for the null space of a matrix.   4.8: 1, 2, 5,   Assignment 9 Exercise 7(a), solutions here
  • Give the matrix of a linear transformation.   5.2: 1, 2, 3b, 5
  • Determine whether a composition of two transformations exists, and give it if it does.   5.3: all

• More Challenging Tests and Justifications:
  • Describe the span of a set of vectors geometrically as a point, line, plane, or all of three-space.   4.1: 1,   Assignment 9 Exercise 1, solutions here
  • Determine whether a given subset of a vector space is a subspace. If it is not, show that it is not.   4.3: 1, 2
  • Determine whether a given set of vectors is a basis for a vector space or subspace.   Assignment 9 Exercise 8, solutions here
  • Give a basis for a vector space or subspace.   4.7: 1, 2, 3
  • Determine whether a transformation is linear. If it is not, show that it is not.   5.2: 1, 2, 4

• Applications:
  • Set up and solve a least-squares problem.   Assignment 8, solutions here.
  • Give matrices, with or without use of homogeneous coordinates as appropriate, that perform transformations in two-space.
  • Give a sequence of matrices, in the proper order, that can be multiplied to obtain a single matrix that gives the results of a sequence of transformations.

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