Honors Linear Algebra and Applications: Syllabus

Objectives are grouped by the student learning outcome they support.

Outcome 1: Operations on vectors

• 1aFind the sum of two vectors and a scalar multiple of a vector.
• 1bProve the basic arithmetic properties of vector operations in Euclidean spaces $R^n$.
• 1cDetermine if a set, together with an addition and a scalar operation, satisfies all the axioms of a general vector space.
• 1dDetermine if a bilinear form on a vector space satisfies all the axioms of an inner product space.
• 1eFind the inner product of two vectors, the angle between two vectors, and the length of a vector.
• 1fDetermine if two vectors are orthogonal.
• 1gApply the Gram-Schmidt algorithm to a basis for an inner product space in order to construct an orthogonal and orthonormal basis for the space.
• 1hFind a basis for the orthogonal complement of a subspace in the finite-dimensional case.

Outcome 2: Systems of linear equations

• 2aConstruct the augmented matrix that represents a system of equations.
• 2bProve that row operations do not change the solution set of a system of equations.
• 2cUse row operations to find the reduced row echelon form (rref) of a matrix.
• 2dApply the Gauss-Jordan algorithm to find the rref of a matrix.
• 2eIdentify the leading and free variables in the rref, and use these to describe all the solutions of the linear system.
• 2fDetermine if a set of vectors is linearly independent or dependent.
• 2gUse the rref to find a basis for the row space, column space, and null space of a matrix, and determine its rank and nullity.
• 2hDetermine if a set of vectors forms a subspace, and if so, find a basis for the subspace and determine its dimension.

Outcome 3: Linear transformations

• 3aDetermine if a function from one vector space to another satisfies the definition of a linear transformation.
• 3bGraph the action of a linear operator on Euclidean 2- or 3-space, including rotations, projections, and reflections.
• 3cPerform arithmetic on matrices, including finding the sum or product of two compatible matrices, the transpose of a matrix, and a scalar multiple of a matrix.
• 3dFind a basis for the kernel and the range of a linear transformation.
• 3eDetermine if a linear transformation is injective (one-to-one) or surjective (onto).
• 3fFind the inverse of a matrix, and the inverse of its associated linear transformation, or explain why the inverse does not exist.
• 3gUse the inverse to solve a matrix equation with a square coefficient matrix.
• 3hConstruct the matrix of a linear transformation from one finite-dimensional vector space to another, relative to a basis for each space.
• 3iConstruct the change-of-basis matrix from one ordered basis to another, in the finite-dimensional case.
• 3jSolve application problems involving linear transformations on abstract vector spaces.

Outcome 4: Eigentheory

• 4aFind the determinant of a square matrix using permutations, row operations, or a cofactor expansion.
• 4bCompute the characteristic polynomial of a square matrix.
• 4cFind the eigenvalues of a square matrix, and a basis of eigenvectors for each of its eigenspaces.
• 4dDetermine if a square matrix is diagonalizable, and if so, apply the diagonalization algorithm.
• 4eProve that a matrix is diagonalizable if and only if it has a complete set of eigenvectors.
• 4fConstruct an orthogonal matrix that diagonalizes a symmetric matrix.

Outcome 5: Proofs

• 5aProve properties of vector and matrix operations, such as the associative property of matrix multiplication.
• 5bProve that certain special sets of matrices, such as diagonal, triangular, and symmetric matrices, are closed under matrix addition, matrix multiplication, and scalar multiplication, and are thus subspaces.
• 5cProve uniqueness properties, such as the uniqueness of: the zero vector; the inverse of an invertible matrix; the coordinate vector of a given vector relative to a fixed basis; the rref of a matrix.
• 5dProve that the image of a set of independent vectors under an injective (one-to-one) transformation is also an independent set.
• 5eProve properties of permutations.
• 5fProve the effects of row and column operations on the determinant of a matrix.
• 5gProve that the determinant function is multiplicative.

1. Euclidean $n$-Space ($R^n$)
   1. Vector arithmetic in $R^n$: addition, scalar multiplication, norm, and linear combinations
   2. Graphing vectors in 2 or 3 dimensions
   3. The span of a set of vectors in $R^n$
   4. The dot product in $R^n$
   5. The Cauchy-Schwarz Inequality
   6. Angle and orthogonality in $R^n$
   7. Systems of linear equations
   8. Elementary row operations on matrices, and their properties
   9. The Gauss-Jordan Algorithm
   10. Linear dependence and independence
   11. The Minimizing Theorem
   12. The Extension Theorem
   13. Subspaces of $R^n$
   14. Basis for a subspace and the dimension of a subspace
   15. The transpose operation
   16. The four fundamental matrix spaces and their relevance:
       1. row space
       2. column space
       3. null space of a matrix
       4. null space of its transpose
   17. The orthogonal complement of a subspace
2. Linear Transformations of Euclidean Spaces
   1. The standard matrix of a linear transformation
   2. Graphing a linear operator
   3. Dilation, contraction, and shear operators in $R^2$
   4. Rotations in $R^2$
   5. Projections and reflections in $R^2$ and $R^3$
   6. Arithmetic operations on linear transformations, and their properties
   7. Compositions of linear transformations
   8. Arithmetic operations on matrices, and their properties
   9. Kernel and range of a linear transformation
   10. Injective transformations and surjective transformations
   11. Applying the Gauss-Jordan Algorithm to find the inverse of an invertible matrix
   12. Properties of invertible matrices
   13. Diagonal, triangular, and symmetric matrices, and their properties
3. Abstract Vector Spaces and Their Linear Transformations
   1. The axioms of an abstract vector space
   2. Examples of abstract vector spaces, including function spaces and matrix spaces
   3. Linear combinations of sets of vectors, including the infinite case
   4. Subspaces of an abstract vector space; basis and dimension
   5. Linear transformations of abstract vector spaces
   6. Coordinate vectors with respect to a fixed basis
   7. The matrix of a linear transformation with respect to two bases
   8. Kernel and range; injective and surjective linear transformations
4. The Determinant Function
   1. Permutations and their properties
   2. The definition of the determinant function using permutations
   3. Using row and column operations to compute the determinant
   4. Cofactor expansion
   5. Cramer’s Rule
5. Eigentheory of a Square Matrix
   1. Definition of eigenvalue and eigenvector, and their geometric interpretation
   2. The characteristic polynomial
   3. Eigenspaces
   4. Applying eigenspaces to diagonalize a matrix, when possible
   5. Change of basis
6. Inner Product Spaces
   1. Axioms of an inner product space
   2. Angles and orthogonality in inner product spaces
   3. Orthogonal and orthonormal sets
   4. The Gram-Schmidt Algorithm
   5. The orthogonal complement of a subspace of an inner product space
   6. Orthogonal decompositions
   7. Projection operators and their matrices
   8. Orthogonal matrices
   9. Orthogonal diagonalization of symmetric matrices
7. Applications of Linear Algebra
   1. Systems of equations in science, such as basic circuit analysis or balancing chemical reactions
   2. The Wronskian of a set of functions
   3. The exponential of a diagonalizable matrix
   4. Finding higher derivatives and integrals of functions in finite-dimensional spaces preserved by the derivative, and extending these ideas to solve linear differential equations with constant coefficients