I have grouped the fundamental ideas of the course into 10 topics. As explained in the syllabus, the bulk of your grade will be determined by how many of these topics you have been certified in. To be certified in a topic, you must be successful on two assessments on that topic. However, different students will take different paths to certification.
For each topic, there will be three written topic assessments available via Gradescope in three consecutive weeks. If you successfully complete two of those, then you're done! If you only successfully complete zero or one of those, each week you can take one new assessment in my student office hours.
For example:
Student X is successful on Topic Assessment 1, version A and, the next week, is successful on T.A. 1, version B. They're certified on Topic 1, so they don't need to take any more assessments on that topic.
Student Y is not yet successful on T.A. 1.A, but is successful on T.A 1.B and T.A. 1.C. They're certified on Topic 1!
Student Z is not yet successful on T.A. 1.A. or T.A. 1.B. However, they are successful on T.A. 1.C. and, since they still need one more successful assessment, make an appointment in office hours to take a new version of T.A. 1. When they're successful on that, they're certified on Topic 1!
Big takeaway: It took Student Z longer than Student X to get certified on Topic 1, but at the end of the day that doesn't really matter--as long as you're eventually certified, you're good to go!
Span and linear independence
I can determine if a vector can be written as a linear combination of a list of vectors.
I can determine if a list of vectors is linearly dependent or linearly independent.
I can determine if a list of vectors spans R^n in multiple ways (e.g. geometrically, algebraically.)
I can relate questions about spanning and linear independence to the existence/ uniqueness of solutions to a system of linear equations or a vector equation.
Solving systems of linear equations
I can translate between a system of linear equations, vector equations, augmented matrices, and geometric representations.
I can compute the solution set for a system of linear equations, and describe it in different ways.
I can explain why a matrix isn't in reduced row echelon form, use row moves to put a matrix in reduced row echelon form, and explain why row moves don't change solution sets.
I can use the reduced row echelon form of a matrix to draw conclusions about when a list of vectors spans or is linearly (in)dependent.
Linear transformations
I can explain whether a map is linear or not, and apply key properties of linear maps.
I can translate between an m x n matrix and a linear map from R^n to R^m.
I can determine whether a linear transformation is one-to-one or onto.
I can relate one-to-one and onto-ness questions to systems of linear equations.
Inverses
I can compute the product of two matrices in multiple different ways, and relate it to the composition of linear maps.
I can determine whether a linear transformation or matrix is invertible.
I can find the inverse of an invertible matrix or transformation.
I can apply the Invertible Matrix Theorem to answer conceptual questions.
Determinants
I can explain how the determinant captures geometric aspects of linear transformations.
I can use row moves to compute the determinant of a matrix.
I can apply properties of the determinant (such as the determinant of a product, determinant of an inverse matrix, determinants of special families like upper triangular or diagonal matrices. )
I can relate the determinant to properties from the Invertible Matrix Theorem.
Abstract vector spaces
I can explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn't a vector space.
I can determine whether a subset of a vector space is a subspace.
I can solve problems involving span or linear independence in general vector spaces.
I can determine whether a map between arbitrary vector spaces is linear.
Special subspaces, bases, and dimension
Given a linear transformation T, I can describe the subspaces Image(T) and Kernel(T), and answer conceptual questions about them.
Given a matrix A, I can compute the subspaces Col(A) and Nul(A), and answer conceptual questions about how they change under row moves.
I can find a basis for a subspace and compute its dimension.
I can apply the rank-nullity theorem to draw conclusions about linear maps and compute the dimension of subspaces.
Eigenstuff
Given a matrix A, I can compute its eigenvalues.
Given a matrix A and an eigenvalue c, I can compute a basis for the eigenspace corresponding to c.
Given a linear transformation T between arbitrary vector spaces, I can describe its eigenvalues and eigenvectors.
I can answer conceptual questions about eigenvalues and eigenvectors.
Coordinates and diagonalization
Given a basis B for a vector space V, I can compute the coordinates of any vector v in V with respect to this basis.
Given a basis B for a vector space V and a linear transformation T from V to V, I can compute the matrix of T with respect to B.
Given a diagonalization of a matrix A, I can identify the eigenvectors and eigenvalues of A.
I can determine whether a matrix is diagonalizable and, if so, diagonalize it.