A set of possible learning outcomes for a Linear Algebra course
Interpret existence and uniqueness of solutions geometrically
Understand algebraic and geometric representations of vectors in R^n and their operations, including addition, scalar multiplication and dot product
Solve systems of linear equations using Gauss-Jordan elimination to reduce to echelon form
Solve systems of linear equations using the inverse of the coefficient matrix when possible
Perform common matrix operations such as addition, scalar multiplication, multiplication, and transposition
Recognize and use equivalent forms to identify matrices and solve linear systems
Discuss associativity and noncommutativity of matrix multiplication
Compute with and recognize properties of particular matrices
Formulate, solve, apply, and interpret properties of linear systems
Perform row operations on a matrix
Find the transpose of a matrix
Recognize and use equivalent statements regarding invertible matrices, pivot positions, and solutions of homogeneous systems
Relate an augmented matrix to a system of linear equations
Relate a matrix to a homogeneous system of linear equations
Multiply matrices
Recognize when two matrices can be multiplied
Qualitative describe features of a matrix, e.g., diagonal, upper or lower triangular
Define what it means for a linear system to be consistent or inconsistent
Determine when a system of linear equations has no, one, or many solutions
Distinguish between homogeneous and nonhomogeneous systems
Identify special matrices like the zero matrix and the identity matrix
Solve linear systems of equations using the language of matrices
Translate word problems into linear equations
Recognize echelon forms
Perform Gaussian elimination
Relate various matrix transformations to geometric illustrations
Define the inverse of a matrix
Compute the inverse of a matrix
List properties of vectors in R^n
Compute an LU decomposition
Provide a definition of the determinant
Use determinants and their interpretation as volumes
Describe how performing row operations affects the determinant
Analyze the determinant of a product algebraically and geometrically
Determine the sign of a permutation
Compute the determinant of a two-by-two matrix
Compute the determinant of a three-by-three matrix
Compute the determinant of a matrix via the formula involving permutations
Compute the determinant of an upper triangular matrix
Compute the determinant of a matrix via cofactor expansion
Describe properties of the determinant
Explain what the determinant measures geometrically
Relate the determinant of three-by-three matrices to the cross product
Describe how the determinant of a matrix and its transpose are related
Describe how the determinant of a matrix and its inverse are related
Use determinants to calculate the inverse of a matrix
Describe how the determinant of a product of matrices relates to the determinant of the individual matrices
Provide an axiomatic description of an abstract vector space
Use axioms for abstract vector spaces (over the real or complex fields) to discuss examples (and non-examples) of abstract vector spaces such as subspaces of the space of all polynomials
Discuss the existence of a basis of an abstract vector space
Describe coordinates of a vector relative to a given basis
Recognize and use basic properties of subspaces and vector spaces
Define subspace of a vector space
List examples of subspaces of a vector space
Determine whether or not particular subsets of a vector space are subspaces
Determine a basis and the dimension of a finite-dimensional space
Discuss spanning sets for vectors in R^n
Discuss linear independence for vectors in R^n
Define the dimension of a vector space
Prove all bases have the same number of elements
Prove elementary theorems concerning rank of a matrix
State the rank-nullity theorem
Prove the rank-nullity theorem
Define the rank of a linear transformation
Define row space and column space of a matrix
Describe a relationship between the row space and column space of a matrix
For a linear transformation between vector spaces, discuss its matrix relative to given bases
Given a linear transformation and bases, find a matrix representation for the linear transformation
Discuss how those matrices change when the bases are changed
Identify properties of a matrix which the same for all matrices representing the same linear transformation
Interpret a matrix as a linear transformation from R^n to R^m
Understand the relationship between a linear transformation and its matrix representation
Describe geometrically significant linear transformations of the plane to itself
Interpret a matrix product as a composition of linear transformations
Interpret the inverse matrix as representing the inverse linear transformation
Distinguish between a matrix as a table of numbers and a linear transformation as a function
Define “injective function”
Define “surjective function”
Define “bijective function”
Identify when a linear transformation is injective
Identify when a linear transformation is surjective
Identify when a linear transformation is bijective
Decide whether a linear transformation is one-to-one or onto and how these questions are related to matrices
Define the image of a linear transformation
Prove that the image of a linear transformation is a subspace
Define the kernel of a linear transformation
Prove that the kernel of a linear transformation is a subspace
Discuss the kernel and image of a linear transformation in terms of nullity and rank of the matrix
Find the eigenvalues and eigenvectors of a matrix
Define eigenvalues and eigenvectors geometrically
Use characteristic polynomials to compute eigenvalues and eigenvectors
Use eigenspaces of matrices, when possible, to diagonalize a matrix
Perform diagonalization of matrices
Explain the significance of eigenvectors and eigenvalues
Find the characteristic polynomial of a matrix
Use eigenvectors to represent a linear transformation with respect to a particularly nice basis
Explain the geometric significance of real versus imaginary eigenvalues for two-by-two matrices
Definitions, examples, and properties
Computational methods
Understand how to determine the angle between vectors and the orthogonality of vectors.
Norms
Orthogonality
Orthogonal bases
Gram-Schmidt orthogonalization
Discuss orthogonal and orthonormal bases
Explain the Gram-Schmidt orthogonalization process
Define orthogonal complements
Define orthogonal projections
Compute the orthogonal projection of a vector onto a subspace, given a basis for the subspace
Discuss rigid motions and orthogonal matrices
Discuss general inner product spaces and symmetric matrices, and associated norms
Explain how orthogonal projections relate to least square approximations