MTH 263 - Introduction to Linear Algebra

1. The student will learn the fundamental properties of matrices.
   1. Add, subtract, multiply, and invert matrices on appropriate occasions and in an appropriate manner.
   2. Describe and solve systems of linear equations with matrices.
   3. Perform elementary row operations with and without elementary matrices.
   4. Define and apply symmetric and skew-symmetric matrices.
   5. Define and apply the determinant of a matrix, and the applications of determinants in a variety of contexts.
   6. Define and apply eigenvalues and eigenvectors.
   7. Recognize digonalizable matrices and transform such matrices into diagonal matrices.
   8. Use the specific properties of symmetric matrices.
   9. Perform orthogonal diagonalization on symmetric matrices.
2. The student will learn the fundamental language and processes of vector spaces and inner product spaces.
   1. Motivate and execute the definitions of vector space and inner-product space.
   2. Recognize vectors, vector spaces, inner-product spaces, and subspaces in a variety of contexts.
   3. Define and apply length and orthogonality in a variety of inner-product spaces.
   4. Define and apply linear dependence/independence and spanning.
   5. Define and apply basis, dimension, and coordinates relative to a basis.
   6. Define and apply an orthonormal basis.
   7. Define and apply the Gram-Schmidt process.
3. The student will learn about linear transformations.
   1. Motivate and execute the definition of linear transformation.
   2. Use the language of linear transformations correctly.
   3. Recognize the consequences of linear transformation on dimensions and bases of vector spaces.
   4. Identify matrices with linear transformations and to represent linear transformations with matrices.
   5. Define and apply the similarity of transformations/matrices.
   6. Employ transition matrices to effect a change of basis.