Portland Community College | Portland, Oregon

This course is designed to familiarize students with the elementary concepts of linear algebra, generally restricting to examples that relate to \(\mathbb{R}^n\). Abstract theory is kept to a minimum. Students are exposed to numerous applications.

Upon completion of this course the student should be able to:

• Analyze real-world scenarios to recognize when use of vectors, matrices, or linear systems is appropriate. 
• Formulate problems, creatively model those problems (using technology, if appropriate), and solve those problems using multiple approaches. 
• Judge if the results are reasonable, and then interpret and clearly communicate those results.
• Work with vectors, matrices, or linear systems symbolically and geometrically. 
• Use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving vectors, matrices, or systems of linear equations with colleagues in the field of mathematics, science or engineering.

Upon completion of this course the learner should be able to:

• Appreciate where linear algebra is encountered in the real world, and be able to communicate the underlying mathematics in order to help another person find similar appreciation.

In-class activities are primarily lecture/discussion and problem-solving sessions. The students may use appropriate technology to investigate and reinforce concepts presented in class.

1. Demonstrate an understanding of various types of linear systems and their applications to real world problems in:

   • At least two in-class proctored exams of one or more hours, one of which is a comprehensive final exam. Proctored exams should be worth at least 50% of the overall grade.

   • At least two of the following measures:

     • Take home exam(s)

     • Quizzes

     • Computer lab assignments

     • ​Homework

2. Demonstrate the ability to communicate with colleagues on the topics of linear algebra by doing:

   • At least one group or individual project with written report and/or oral in-class presentation

   • At least one of the following:

     • Participation in class discussions.

     • In-class group activities

     • Attendance

1. Context Specific Skills
   • Students will learn to describe linear structures verbally, geometrically, symbolically, and numerically.
   • Students will learn to recognize underlying vector space structures in a variety of abstract and applied contexts.
   • Students will learn to apply the terminology and notation of Linear Algebra correctly and appropriately in a variety of abstract and applied contexts.
   • Students will learn to compute the following matrix calculations by hand for at least \(3 \times 3\) matrices: row echelon form, reduced row echelon form, matrix inverse, and a variety of arithmetic operations.
   • Students will learn to construct linear models for a variety of applied problems.
   • Students will acquire proficiency in the use of linear transformations of \(\mathbb{R}^n\) and matrix algebra in solving a variety of abstract and applied problems.
2. Learning Process Skills
   • Classroom activities will include lecture/discussion and group work.
   • Students will communicate their results in oral and written form.
   • Students will apply concepts to real-world problems.
   • Calculators and/or computers will be used by the students for tasks such as row reduction, diagonalization, special factorization of matrices, and solving systems of linear equations.

1. Matrix Algebra
   1. Systems of Linear Equations.
      1. Write the augmented matrix of a system of \(m\) linear equations in \(n\) variables.
      2. Identify matrices that are in row echelon form and reduced row echelon form.
      3. Transform an \(m \times n\) matrix to reduced row echelon form using elementary row operations and Gauss-Jordan elimination.
      4. Solve a system of \(m\) linear equations in \(n\) variables by transforming its augmented matrix to reduced row echelon form.
      5. Identify when a linear system is consistent and when it is inconsistent and interpret the solution geometrically.
      6. For consistent systems, identify when the system has one unique solution and when it has infinitely many solutions.
      7. When a system has infinitely many solutions, express the solution set by solving for the pivot variables in terms of the free variables. Write the solution set in parametric form and in vector form.
      8. Define a homogeneous linear system and recognize that homogeneous systems are always consistent. Distinguish between the trivial solution and non-trivial solutions and discuss ways to identify when a given homogeneous system has non-trivial solutions.
      9. Recognize that the solution set to a non-homogeneous system is a shift of the solution set for the homogeneous system that has the same coefficient matrix.
   2. Matrix Operations.
      1. Define the operations of addition and scalar multiplication of \(m \times n\) matrices. Study the algebraic properties of \(m \times n\) matrices under these operations.
      2. Define the transpose of an \(m \times n\) matrix and demonstrate the ability to use transpose properties (sums, scalar products, products, transpose of a transpose, etc.).
      3. Define the product of an \(m \times n\) matrix and an \(n \times p\) matrix. Study the algebraic properties of matrix multiplication.
      4. Define the \(n \times n\) identity matrix \(I\). Study the identity property of matrix multiplication.
      5. Define and study the properties of invertible matrices. Compute the inverse of an invertible matrix and identify when a square matrix is not invertible.
      6. Write a system of linear equations as a matrix equation. If the coefficient matrix of the system is invertible, solve the system using the inverse of the coefficient matrix.
      7. Define the determinant of a square matrix and compute determinants by cofactor expansion across any row or down any column of a square matrix. Use determinants to determine the invertibility of a square matrix.
2. \(n\)-Dimensional Euclidean Space, \(\mathbb{R}^{n}\)
   1. Define addition and scalar multiplication of vectors in \(\mathbb{R}^{n}\). Study the algebraic properties of vectors under vector addition and scalar multiplication.
   2. Define the linear dependence or independence of a subset of vectors in \(\mathbb{R}^{n}\). Determine the linear dependence or independence of a given set of vectors by solving a homogeneous linear system.
   3. Define a linear combination of vectors in \(\mathbb{R}^{n}\). Solve a linear system to determine if a particular vector is a linear combination of a given set of vectors.
   4. Recognize that a square matrix is invertible if and only if its column vectors are linearly independent.
   5. Define a subspace of \(\mathbb{R}^{n}\). Determine whether a given subset of \(\mathbb{R}^{n}\) is a subspace.
   6. Define the span of a set of vectors from \(\mathbb{R}^{n}\) and recognize that the span of a subset of vectors from \(\mathbb{R}^{n}\) is a subspace of \(\mathbb{R}^{n}\).
   7. Define the null space and the column space of an \(m \times n\) matrix \(A\) and study properties and the significance of these subspaces.
   8. Define a basis for \(\mathbb{R}^{n}\) or a subspace of \(\mathbb{R}^{n}\). Determine whether a given subset of vectors from \(\mathbb{R}^{n}\) forms a basis for \(\mathbb{R}^{n}\). Determine whether a given subset of vectors from a subspace of \(\mathbb{R}^{n}\) forms a basis for that subspace.
   9. Define the dimension of a subspace of \(\mathbb{R}^{n}\). Study the relationships between linearly independent sets, spanning sets, bases, and dimension. Extract a basis from a set of vectors that spans \(\mathbb{R}^{n}\) or a subspace of \(\mathbb{R}^{n}\). Extend a linearly independent subset of vectors to a basis for \(\mathbb{R}^{n}\) or a subspace of \(\mathbb{R}^{n}\).
   10. Define an ordered basis for \(\mathbb{R}^{n}\). Define the coordinates of a particular vector with respect to a given ordered basis. Compute the coordinates of a vector with respect to a given ordered basis. Use a transition matrix to convert the coordinates for a vector with respect to one ordered basis to its coordinates with respect to a second ordered basis.
3. Orthogonality
   1. Define the dot product on \(\mathbb{R}^{n}\). Study algebraic properties of the dot product.
   2. Define distance and magnitude in terms of the dot product. Study algebraic properties of distance and magnitude.
   3. Define orthogonal vectors. Define orthogonal and orthonormal subsets of \(\mathbb{R}^{n}\).
   4. Use dot products to compute the coefficients of vectors in \(\mathbb{R}^{n}\) with respect to orthogonal and orthonormal bases.
   5. Determine the orthogonal decomposition of a vector relative to a subspace of \(\mathbb{R}^{n}\) and the orthogonal complement of that subspace of \(\mathbb{R}^{n}\).
   6. Use the Gram-Schmidt process to produce an orthogonal basis for a given subspace of \(\mathbb{R}^{n}\).
   7. Decompose an invertible square matrix \(A\) into a product \(QR\), for \(Q\) an orthogonal matrix, and \(Q'R'\), for \(Q'\) an orthonormal matrix. In this case, \(R' = (Q')^{T}A\).
4. Linear Transformations
   1. Properties of Linear Transformations.
      1. Define a linear transformation from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m}\). Distinguish between linear and non-linear transformations. Study algebraic properties of linear transformations.
      2. Define the null space (kernel) and the range (image) of a linear transformation \(T : \mathbb{R}^{n} \to \mathbb{R}^{m}\). If \(\operatorname{N}(T)\) is the null space of \(T\), then \(\operatorname{N}(T)\) is a subspace of \(\mathbb{R}^{n}\). If \(\operatorname{R}(T)\) is the range of \(T\), then \(\operatorname{R}(T)\) is a subspace of \(\mathbb{R}^{m}\).
      3. Define the nullity and the rank of a linear transformation. Study the rank-nullity theorem; i.e. if \(T : \mathbb{R}^{n} \to \mathbb{R}^{m}\) is a linear transformation, then \(\dim(\operatorname{R}(T))+\dim(\operatorname{N}(T)) = n\).
   2. Matrix Representations of Linear Transformations.
      1. Define the matrix representation of a linear transformation \(T : \mathbb{R}^{n} \to \mathbb{R}^{m}\) relative to the standard bases for \(\mathbb{R}^{n}\) and \(\mathbb{R}^{m}\). Compute the matrix representation of such a linear transformation.
      2. Define the matrix representation of a linear transformation \(T : \mathbb{R}^{n} \to \mathbb{R}^{m}\) relative to arbitrary ordered bases, \(B_{1}\) and \(B_{2}\), for \(\mathbb{R}^{n}\) and \(\mathbb{R}^{m}\), respectively. Compute the matrix representation of such a linear transformation relative to \(B_{1}\) and \(B_{2}\).
5. Eigenvalues and Eigenvectors
   1. Define eigenvectors and eigenvalues of an \(n \times n\) matrix, \(A\). Calculate eigenvalues by solving the characteristic equation of the matrix, \(\det(A - \lambda I) =0\). Calculate eigenvectors by solving the linear system \((A - \lambda I)\vec{x} = \vec{0}\).
   2. Define the eigenspace corresponding to an eigenvalue \(\lambda\). Prove that an eigenspace is a subspace of \(\mathbb{R}^{n}\).
   3. Determine geometric and algebraic multiplicities for all eigenvalues of a given \(n \times n\) matrix \(A\). The geometric multiplicity of \(\lambda\) equals the nullity of \((A - \lambda I)\).
   4. Emphasize the significance of the eigenvalue \(\lambda = 0\), as it relates to the invertibility of \(A\).
   5. Define a diagonalizable \(n \times n\) matrix \(A\). Define an eigenbasis for \(A\). Prove an \(n \times n\) matrix \(A\) is diagonalizable if and only if \(\mathbb{R}^{n}\) has a basis consisting of eigenvectors of \(A\). Diagonalize a diagonalizable \(n \times n\) matrix.
   6. Compute powers of a diagonalizable matrix; i.e. if \(A = PDP^{-1}\), then \(A^{n} = PD^{n}P^{-1}\).
6. Additional Topics and Applications
   1. The course will cover three or more linear algebra topics and applications similar in depth to those listed below.
      1. General Vector Spaces: Define a vector space in terms of an arbitrary set with defined operations of vector addition and scalar multiplication that satisfy the vector space axioms. Introduce different examples of vector spaces including matrix spaces and function spaces, under different definitions of vector addition and scalar multiplication.
      2. Applications of matrix powers in linear recursion relations.
      3. Two-dimensional computer graphics.
      4. Three-dimensional computer graphics with perspective.
      5. Systems of differential equations.
      6. Second derivative test in two dimensions or higher (Hessian matrix).
      7. Principal component analysis of statistical data (from the covariance matrix).
      8. Principal axes of rotation (from moment of inertia matrix).
      9. Principal axes for quadric surfaces.
      10. Transforms (\(z\), Fast Fourier, Walsh) and filters for digital signal processing.
      11. Fourier series approximations of periodic functions.
      12. Least squares approximation (more general context than statistics).
      13. Path components of digraphs (via powers of the adjacency matrix).
      14. Markov chains (powers of transition matrices).
      15. Linearization of non-linear systems of equations.
      16. Encryption and coding of messages and other data.