Linear outcomes for Linear Algebra

A set of possible learning outcomes for a Linear Algebra course

Matrices

  • 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

Determinants

  • 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

Vector spaces

Linear independence

Basis and dimension

  • 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

Linear transformations

  • 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

Eigenvalues and eigenvectors

  • 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

Inner products

  • 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