This semester I’m teaching our introductory linear algebra course. As I did for Calculus II, I’ve implemented a standards-based assessment system. I’ve taken our course content and split it into “standards”, or little pieces of mathematics that I want my students to master. These standards are grouped together by what I call “Big Questions”. Here is what we’ve covered so far this semester:

• Big Question #1: What are the tools for solving systems of linear equations?
  • 1.1: I can solve systems of linear equations using row operations. I can use Gaussian elimination with back-substitution to solve systems of linear equations. I can use Gauss-Jordan elimination to solve systems of linear equations.
  • 1.2: I can characterize the solutions to systems of linear equations using appropriate notation and vocabulary.
  • 1.3: I can use matrix inverses to solve systems of linear equations.
  • 1.4: I can find and use an LU-factorization of a matrix to solve a system of linear equations.
• Big Question #2: What is the fundamental structure of the algebra of matrices?
  • 2.1: I can perform algebraic operations with matrices, including addition, subtraction, scalar multiplication, and matrix multiplication. I can compute the transpose of matrices.
  • 2.2: I can find the inverse of matrices using Gaussian elimination. I can find the inverse of matrices using a product of elementary matrices.
  • 2.3: I can demonstrate theoretical connections about properties in the algebra of matrices.
• Big Question #3: How can we characterize invertible matrices?
  • 3.1: I can find determinants using cofactor expansion. I can find determinants using row or column operations.
  • 3.2: I can demonstrate theoretical connections between statements equivalent to “the matrix A is invertible.”
  • 3.3: I can demonstrate theoretical connections between matrix equations, vector equations, and systems of linear equations, and their properties and solutions.
• Big Question #4: What are vector spaces & how can we describe them?
  • 4.1: I can prove whether an algebraic structure is a vector space (or not) using the vector space axioms. I can prove whether or not a subset W of a vector space V forms a subspace. I can determine and characterize subspaces of $\mathbb{R}^n$.
  • 4.2: I can write a proof showing whether a subset of vectors from a vector space forms a spanning set for the vector space (or not). I can write a proof to show whether a subset of vectors from a vector space is linearly independent (or not). I can determine whether a set of vectors forms a basis for a vector space. I can find the dimension of a vector space.
  • 4.3: I can find a basis for the row space, the column space, or the null space of a matrix. I can determine the rank and nullity of a matrix. Given a consistent system Ax=b, I can describe the general solution in the form x=x_p+x_h
  • 4.4: I can demonstrate knowledge of the theory of vector spaces by proving elementary results and theorems.

The remaining Big Questions are:

• Big Question #5: What are inner product spaces and how can we describe them?
• Big Question #6: What kinds of functions map one vector space into another while preserving vector space operations?
• Big Question #7: What are eigenvalues and why are they useful?

Our first exam was last week, so today has been Re-Assessment Central in my office. I’ll hand back our exams tomorrow and I’m hoping to talk with my students more about standards-based grading and how they can improve their standing in the course.