When the English mathematician James Joseph Sylvester (1814-1897) first used the term matrix in 1850, he probably had little idea what would develop from this simple concept!
Of course, the word had been in use for millenia, having been drawn directly from the Latin name for a pregnant animal (from mater = mother). By the later 14th century, it had come to mean uterus/womb. By the mid sixteenth century it had started to take on the meaing of a place where something develops. For James Joseph Sylvester, the something was a determinant. He viewed a matrix as an array of numbers which could generate smaller square arrays of numbers (called determinants). So, a matrix was the ‘mother’ out of which determinants were born!
It was Sylvester’s friend Arthur Cayley (1821-1895) who first treated matrices as mathematical objects in their own right and began to develop an algebra of matrices in the late 1850s. Mathematicians raced to discover their properties. Could they be added and subtracted? What about multiplied and divided? Were any of the operations commutative or associative (i.e. did it matter what order you performed the operations)? Were some matrices equivalent to others? Were there identity matrices? The list of questions grew … as did mathematicians’ understanding of these amazing objects.
They are used to solve complex arrays of simultaneous equations, as operators to transform graphs and images in a vast number of ways (rotations, reflections, stretches, etc.), to simplify our understanding of vectors, tensors and geodesics in physics … and the list goes on! When you modify images using graphics software, or move through a landscape in a computer game, the operations that govern the video output are essentially matrix operations.
This is a rich and interesting field of study.
Please watch this space. I will add material as soon as I am able.
Thanks for this video, I was having a lot of trouble with the chain rule, I studied the text book (very confusing) and several other videos, but I found your video most helpful, and I new feel I really do understand it.
MeijinSensei (on a CCM YouTube video about the Chain Rule)
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