MATH 208 Introduction to Linear Algebra • 5 Cr.
Introduces the vocabulary, algebra, and geometry of vector spaces in "R" and function spaces. Students use matrix methods and vectors to explore systems of linear equations and transformations. Also presents elementary theory of eigenvalues. Fulfills the quantitative or symbolic reasoning course requirement at BC. Recommended: MATH& 153.
After completing this class, students should be able to:
- OuSolve systems of linear equations -- small ones manually by row reduction techniques and larger ones using technology.
- Use linear systems to model and analyze applied situations.
- Perform matrix operations, including matrix inversion.
- Translate linear systems into matrix equations, and use matrix inverses to solve, where appropriate.
- Perform vector operations in Rn?and interpret them geometrically in R? and R?.
- Use vectors to solve "physical" problems.
- Verify and/or refute the validity of vector space axioms in specific examples.
- Use the vocabulary of vector spaces (linear combination, span, subspace, linear independence and linear dependence, basis, dimension, and orthogonal) appropriately in R? and R?.
- Apply the vocabulary of vector spaces (linear combination, span, subspace, linear independence and linear dependence, basis, dimension, and orthogonal) to specific examples in Rn.
- Identify and construct examples of linear combinations, spans, subspaces, linear independence and linear dependence, bases, dimension, and orthogonality in spaces of matrices and function spaces.
- Exemplify linear transformations in R?, R??and more general settings, and distinguish linear transformation from non-linear mappings.
- Construct and analyze matrix representations of linear transformations, and relate them to matrix operations.
- Describe the effect (including null space and image) of linear transformations on sets of vectors, especially in R? and R?.
- Compute determinants of 2x2 and 3x3 matrices, and relate them to the geometry of linear transformations and to the solution of linear systems.
- Compute eigenvalues and eigenvectors, interpret them geometrically, and use them to help analyze some applied situation.
- Write simple two or three-step proofs, and construct illustrative counterexamples.