![]() ![]() The matrix that describes this mapping-of-a-mapping is the one you get by multiplying the matrix of the first mapping and the matrix of the second mapping together. We can then use a mapping again and find the image of the image of that domain. That is, we can start with a domain and a mapping, and find the image of that domain. Or the way we can approximate a complicated surface with a bunch of triangular plates. ![]() We do this in the same way, and for pretty much the same reason, we can approximate a real and complicated curve with a bunch of straight lines. We can write a matrix that approximates the original mapping, at least in some areas. A mapping might follow different rules in different regions, but that’s all right. These are ones that turn the domain into the range by stretching or squeezing down or rotating the whole domain the same amount. Properly, a matrix made up of real numbers can only describe what are called linear mappings. A matrix can describe how points in a domain map to points in a range. One of the big uses of matrices is to represent a mapping. We give that stuff wonderful names like traces and determinants and eigenvalues and eigenvectors and such. And for square matrices, ones with equal numbers of rows and columns, we can find other useful stuff. The definition looks like a lot of work, but it represents something useful that way. We can define multiplication, at least sometimes. We can multiply them by real- or complex-valued numbers, called scalars. ![]() (I’m sure somebody, somewhere has created matrices with something else as elements. When they’re not real numbers they’re complex-valued numbers. Inside each individual row and column is some mathematical entity. What we mean by a matrix is a collection of some number of rows and columns. So for them, the matrix was the source of something else. The first mathematicians to use the word “matrix” were interested in things derived from the matrix. History has outvoted him, and his preferred “block”. The word meant the place where something was developed, the source of something else. “Surely,” he wrote, “ means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. And it isn’t that he disliked matrices particularly. The request comes from Gaurish, chief author of the Gaurish4Math blog. Today’s Leap Day Mathematics A To Z term is a famous one, and one that I remember terrifying me in the earliest days of high school. I get to start this week with another request. ![]()
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