Math 330

Click on the links below for sets of review problems and other review materials. You will need to use the Acrobat Reader to view and print these problem sets. If you are using a computer in one of the student labs on campus, this should already be installed and should activate automatically. If your computer does not have the Acrobat Reader, you may download it by clicking here.

Exam 1: (sections 1.1 to 1.9)

• Review Problems with solutions
  
• Be ready for "True/False" and "Explain..." problems similar to those we covered in class.
• Know the definition of linear independence.

• Review Problems with solutions
  
• As with exam 1, be prepared for "True/False" and "Explain..." questions -- I've included a few "explain" questions on the review problems.
• Be able to give definitions for the following terms:
  • Nul A and Col A (4.2)
  • kernel and range of a linear transformation (4.2)
  • linearly independent set of vectors in a vector space (4.3)
  • basis for a vector space (4.4)
  • coordinates of a vector x relative to a basis (4.4)
• Also, I may ask you to prove one of the following things:
  • (j) implies (d) from the IMT
  • (d) implies (c) from the IMT
  • If A is an m by n matrix then nulA is a subspace of Rⁿ. (See Theorem 2 on p.199.)
  • If {b₁,b₂,...,b_n} is a basis for a vector space V and x is any vector in V, then there is a unique set of scalars c₁,c₂,...,c_n so that x = c₁b₁ + c₂b₂ + ... + c_nb_n. (See Theorem 7 on p.216.)

• Exam 3 Review Problems with solutions
• As always, there will be "True/False" and "Explain..." questions.
• Be able to give definitions for the following terms:
  • dimension of a vector space
  • rank of a matrix
  • row space of a matrix
  • eigenvector, eigenvalue, and eigenspace
  • orthogonal set, orthonormal set (6.2)
• Also, I may ask you to prove one of the following things:
  • If there is a basis for Rⁿ consisting of eigenvectors for a matrix A then A = PDP^-1 where P is the matrix with columns equal to the basis of eigenvectors and D is a diagonal matrix with the corresponding eigenvalues on the diagonal. (Half of Theorem 5 on p.282.)
  • Every orthogonal set of nonzero vectors is linearly independent. (Theorem 4 on p.338.)

• The final exam is comprehensive in that it will include topics from the entire semester. However, I will not include any questions from the following sections of the text:
  • 1.6 -- applications of linear systems
  • 3.3 -- Cramer's rule
  • 4.7 -- Change of basis
  • 5.4 -- Eigenvalues in linear transformations
• With the exception of the above listed sections, you should expect exam problems to be similar to homework problems from the sections we covered.
• Sections 6.5 and 7.1 (covered since exam 3) will definitely appear on the final exam.
• There will be some True/False questions, as with our other exams.
• I may ask you one of the following definitions -- the most important concepts we have covered:
  • linearly independent set of vectors
  • Nul A
  • a basis for a vector space
  • an eigenvector for A corresponding to an eigenvalue
  • orthogonal set of vectors
• There will not be a proof on the final exam.

• Note: the exams below were from a summer term version of Math 330, so the structure was a little different and the time a little more limited. There was no cumulative final exam in the summer, the exam breaks were at slightly different places, and there are a few sections we will cover in this course that were not included in the summer class. Still, these exams should give you some idea of the type of problems I put on exams.
• Exam 1 (roughly covers 1.1 through 2.2) with answers
• Exam 2 (roughly covers 2.3 through 4.4, but leaves off 3.3) with answers
• Exam 3 (roughly covers 4.5 through 6.4) with answers