Linear Algebra--Review for Exam 2
(Math 3191, Section 002, Fall 2005)

The exam will cover the material in the following sections: 2.3,3.1,3.2,4.1-4.8.   In the following review, the material for the course is broken down into three levels: "C"-level, "B"-level, and "A"-level. Roughly, 70% of the points on the test will be based on the "C"-level material, 20% from the "B" level material, and 10% from the "A"-level

"C"-level material

(This section will make up 70% of the points on the exam).

Fundamental Skills and Concepts

  1. Calculate the determinant of a 2 by 2 matrix
  2. Invert a 2 by 2 matrix
  3. The invertible matrix theorem.  Possible  questions:
    1. True/False
    2. Write a statement that is equivalent to A being invertible which involves each of the following concepts:
  4. Calculate the determinant by cofactor expansion.  (Make sure you get the sign right for cofactors).
  5. Relationship between determinant and invertibility.
  6. Calculate the determinant of a triangular matrix.
  7. Determine whether a set is a subspace of a vector space.
  8. Determine a basis for the null or column space of a given matrix.
  9. Determine when a set of vectors forms a basis.
  10. Given the coordinates of a vector relative to a given basis in R^n, determine what that vector is.
  11. Determine the coordinates of a vector in R^n relative to a given basis.
  12. Determine the rank of a matrix.
  13. Given the rank of an m x n matrix, determine the dimensions of the null space, the row space, the column space and the null space of A^T  (using the rank theorem).
  14. Determine whether or not a given vector is in the null space of a given matrix.
  15. Relationship between null space and 1-1.
  16. Relationship between column space and onto.

"B"-level material

(This section will make up roughly 20% of the points on the exam). The following is a summary of the types of things you should be able to do (in addition to the "C"-level material) to get a B on the exam:

Key skills and concepts

  1. Calculate the determinant of a matrix by row reduction.
  2. Know what effect each of the elementary row operations has on the determinant.
  3. Calculate the determinant of the product of two square matrices (given the determinant of each of the matrices).
  4. Determine the kernel and range of a linear transformation.
  5. Express a subspace as the span of a given set of vectors.
  6. Construct a basis from a spanning set by discarding unneeded vectors.
  7. Construct a basis from a linearly independent set of vectors by adding vectors.
  8. Determine the change of coordinates matrix from a given basis to the standard basis.
  9. Find the dimension of a given vector space.
  10. Prove that a certain vector space property is or isn't satisfied for a given set and specified operators.
  11. Determine when a set of vectors in a vector space is linearly independent (by using coordinate vectors).
  12. Determine when a set of vectors in a vector space spans the space (also using coordinate vectors).

"A"-level material

(This section covers at most 10% of the points on the exam).

The "A"-level material covers ALL the material assigned in the study guides. Additionally, you must be able to demonstrate a superior understanding of the theoretical material. For example, you should be able to do the following:

  1. Prove that a certain set along with corresponding operators is (or isn't) a vector space.
  2. Prove certain properties of vector spaces, using only axioms of vector spaces. 
  3. Given the coordinates of a vector relative to a basis, determine the coordinates of the vector relative to a different basis.
  4. Determine the change of coordinates matrix from a given basis to another (nonstandard) basis.
  5. Thoroughly understand the Invertible Matrix Theorem, including any additions to it covered in Chapters 3 and 4.
  6. Determine the coordinates of a vector in a general vector space relative to a given basis.
  7. Do any of the "A"-level problems assigned in the study guides.