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