Advanced Linear Algebra - Advanced Linear Algebra, Lecture 1.3: Direct sums and products

Advanced Linear Algebra, Lecture 1.3: Direct sums and products The complement of a subspace Y of X is any subspace Z such that every vector in x can be written uniquely as x=y+z, a sum of elements in Y and Z. In this case, we say that X is a direct sum of Y and Z, and write X=Y⊕Z. Another way to "multiply" two vector spaces to get a larger space is by taking a direct product, which is a new vector consisting of all ordered pairs (y,z), for y in Y and z in Z. This is denoted by Y⊕Z. Though Y⊕Z and Y x Z are isomorphic, this need not hold when there are infinitely many factors, and we see why the direct sum of countably many copies of the real numbers is "smaller" than the direct product. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html