STANDARD COMPETENCIES:

 

 1.       Solve linear systems using the elimination method, augmented matrices, inverse matrices, and determinants.

 2.       Perform matrix operations in multiple dimensions including addition, subtraction, multiplication, row reduction, finding inverses, and calculating determinants.

 3.       Find equations and solve problems involving lines and planes in 3 dimensional space.

 4.       Recognize, provide examples of, and understand the properties of vector spaces.

 5.       Find subspaces, dimensions, and bases for vector spaces.

 6.       Determine linear independence for a set of vectors and test vectors for spanning.

 7.       Solve problems involving inner product spaces and find orthonormal bases for inner product spaces.

 8.       Define a linear transformation.

 9.       Find the kernel and range of a linear transformation, find and apply the matrix of transformation from one basis to another, compose linear transformations, and find coordinates of a vector with respect to a give basis.

 10.         Define an eigenvalue and its eigenvector and find eigenvalues and eigenvectors of a linear transformation.

 11.         Show that a given matrix is similar to another and find a diagonal matrix that is similar to the matrix of a linear transformation.

 

 TOPICAL OUTLINE:

 

 I.      Equations and Matrices: Solutions of Linear Systems, Matrix Arithmetic,

 II.     Matrix Solution Approaches, Determinants, and Cramer`s Rule.

 III.    Vectors:  Vector Spaces, Vector Arithmetic, Lines and Planes in 3-Space,

 IV.     General Properties of Vector Spaces, Subspaces, Basis and Dimension of Spaces, and Coordinates.

 V.      Linear Transformations:  General Characteristics of Transformations,

 VI.     Matrices of Transformation, and Similarity of Matrices.

 VII.    Eigenvalues: Eigenvalues and Eigenvectors, and Diagonalization of Matrices.