INSIDE AN ABSTRACT VECTOR

In engineering, a vector has three scalar components which correspond to the three dimensions of the space in which we live. In least-squares data analysis, a vector is a one-dimensional array that can contain many different things. Such an array is an ``abstract vector.'' For example, in earthquake studies, the vector might contain the time an earthquake began as well as its latitude, longitude, and depth. Alternately, the abstract vector might contain as many components as there are seismometers, and each component might be the onset time of an earthquake. In signal analysis, the vector might contain the values of a signal at successive instants in time or, alternately, a collection of signals. These signals might be ``multiplexed'' (interlaced) or ``demultiplexed'' (all of each signal preceding the next). In image analysis, the one-dimensional array might contain an image, which could itself be thought of as an array of signals. Vectors, including abstract vectors, are usually denoted by boldface letters such as $\bold p$ and $\bold s$.Like physical vectors, abstract vectors are orthogonal when their dot product vanishes: $\bold p \cdot \bold s =0$.Orthogonal vectors are well known in physical space; we will also encounter them in abstract vector space.