Applied Linear Algebra

Upon completion of MATH 2210, the successful student will be able to:

• Perform basic arithmetic computations with vectors in Euclidean two-space, R², and three-space, R³ and generalize these computations to Euclidean n-space, Rⁿ
• Define geometric aspects of vectors in R² and R³ such as length, the dot product, the angle and distance between two vectors, and orthogonality, and generalize these definitions to Rⁿ
• Solve problems using the various forms of the equations of lines and planes in R³
• Express a system of linear equations in vector form or matrix form, and convert between each form
• Solve systems of linear equations using Gaussian elimination and row reduction
• Describe properties of solutions to homogeneous systems of linear equations, and the connection between solutions of homogeneous and non-homogeneous linear systems
• Give a geometric interpretation of subspaces of R² and R³ and generalize these interpretations to Rⁿ
• Define span and linear independence for a set of vectors, and determine if a set of vectors in Rⁿ is linearly independent
• Use the concepts of subspaces, linear independence and span to give a geometric interpretation of solutions to systems of linear equations
• Define basis and dimension, and determine if a set of vectors is a basis for a subspace of Rⁿ
• Perform basic operations on matrices: addition, scalar multiplication, matrix multiplication, transpose, powers, and apply appropriate properties of matrix algebra when performing these operations
• Define a matrix inverse and apply properties of matrix inverses
• Determine the inverse of a matrix by Gaussian elimination
• Define and determine the rank of a matrix
• Determine the basis and dimension for the fundamental subspaces of a given matrix
• Define a linear transformation and determine if a given transformation is a linear transformation
• Interpret linear transformations in terms of matrices and determine the standard matrix for a linear transformation from Rⁿ to R^m
• Determine the matrices that describe a rotation, a shear, a dilation or contraction, and a reflection in R²; Given a 2 x 2 matrix, describe the transformation in terms of the foregoing matrices
• Determine if a linear transformation is one-to-one, onto or invertible
• Determine the kernel, range, rank and nullity of a linear transformation and interpret these in terms of the fundamental subspaces of a matrix
• Form composite transformations from given linear transformations
• Determine the coordinates of a vector in a given basis and determine the transition matrix for a change of basis
• Evaluate the determinant of an n X n matrix
• Apply basic properties of determinants when evaluating the determinant of a matrix
• Discuss the solvability of a system of linear equations using determinants
• Define a complex number, a complex conjugate and the complex plane
• Perform basic arithmetic computations with complex numbers
• Express complex numbers in polar form, and work with DeMoivre’s formula and Euler’s formula
• Define and give a geometric interpretation of eigenvalues and eigenvectors of a matrix
• Determine the characteristic polynomial, eigenvalues and corresponding eigenspaces of a given matrix, including those involving complex numbers
• Define similar matrices and use this property to diagonalize a square matrix
• Define standard basis, orthogonal basis, orthonormal basis, orthogonal projections, orthogonal matrix, orthogonal complement
• Calculate the projection of one vector onto another in Rⁿ
• Convert a basis into an orthonormal basis using the Gram-Schmidt Process (max of three vectors) in Rⁿ
• Orthogonally diagonalize a symmetric matrix
• Use orthogonal projection to find the least-squares solution of a system of linear equations
• Solve application problems related, but not limited, to: electrical network analysis, vector differential equations, dynamical system computer graphics, network analysis traffic flow