Macalester College - Fall 2013
A detailed syllabus is available at http://danflath.com/math-236-fall-2013/.
Linear algebra is one of the pillars of modern mathematics, both pure and applied. It is as important as calculus, maybe more important. You will encounter it and learn more about it in virtually every advanced math course you ever take. This course is just a beginning.
Linear algebra is about what you can do with linear combinations, which are simply sums with coefficients. You are already familiar with examples, such as polynomials: 3 x^2 + 2 x + 5. You may have encountered vectors in space such as 4 i + 3 j + 2 k.
In linear algebra as in every subject in life there are both objects and interactions between objects that require attention. The objects are vector spaces (which are collections of vectors) and the interactions are provided by linear transformations (which are functions that take vectors as inputs and produce vectors as outputs – you can think of matrices).
In linear algebra both concepts and computations are important. The concepts are not difficult, but there are a lot of them. There is a good deal of essential vocabulary. Here is a short list of some of the most important terms: linear combination, linearly independent/dependent, span, basis, dimension, columnspace, rank, nullspace, coordinates with respect to a basis, linear transformation, matrix of a linear transformation, determinant, eigenvector, eigenvalue, inner product, orthonormal basis, projection, orthogonal transformation. For computations, we get some assistance from computers, especially Mathematica.
Vector spaces, bases, dimension
Linear transformations, matrix of a linear transformation
Good bases in presence of inner products: orthonormal bases
Good bases in presence of linear transformation: bases of eigenvectors
Symmetric matrices; singular value decomposition
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