In order to describe and solve the basic linear algebra problems of this chapter, we need some notation. We will refer frequently to matrices, vectors and scalars. A matrix will be denoted by an upper case letter like A, and its -th element by . Occasionally in detailed algorithmic descriptions we will instead write . The submatrix of A occupying rows i though j and columns k through l will be denoted . A lower case letter like x will denote a vector, and its i-th element will be written . Vectors will almost always be column vectors, which are the same as matrices with one column. Lower case Greek letters (and occasionally lower case letters) will denote scalars. will denote the set of real numbers, the set of n-dimensional real vectors, and the set of m-by-n real matrices. will denote the transpose of the matrix A: . A matrix is called symmetric if . A matrix A is positive definite if for all nonzero vectors x. Other notation will be introduced as needed.