# 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

#### 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