Linear Equations
Linear equations can be expressed as , where is the unknown, is a given vector, and is a matrix. The set of solutions is an affine set. In turn, any affine set can be described as the set of solutions to an affine equation. The issue of existence and unicity of solutions lead to important notions attached to the associated matrix . The nullspace, which contains the input vectors that are crushed to zero by the associated linear map ; the range, which contains the set of output vectors that is attainable by the linear map; and its dimension, the rank. There is a variety of solution methods for linear equations; we describe how the QR decomposition of a matrix can be used in this context. |