If u and v are two column vectors, then is defined to be, where is the row vector obtained by taking the “transpose” of u. Since we can write any vector in the plane as a column vector, this gives us a first hint at how we might multiply boxes of numbers together. If and, then, and we get the nice property that the length of u is. You can’t just multiply vectors together (our intuition of multiplying numbers together sort of fails us when we’re dealing with numbers that have direction), but there are two natural(ish) sorts of products you can do. Given two vectors u and v, there’s this thing called the dot product that you can do.
Since a vector is just the arrow, not attached to any particular location, you can move it around (without changing length or direction) and still call it the same thing, so could also be the vector that starts at (1, 1) and goes up 1 and over 2 to the point (2, 3). If you understand what the coordinate (1,2) is, then replace the ( ) with and you’ve got the vector which starts at the origin and ends at the point (1, 2).
#Lattice math how to#
Okay, so last time we saw that vectors were arrows, but I didn’t say how to write them down. Initially, I wanted to motivate all that is The Matrix, but that turned out to be a super huge ordeal that I decided was ultimately not worth it. We can say is an element of the set of 2×2 matrices, or an element of the set of invertible linear transformations from the plane to itself. It could be the matrix of the coefficients of the left-hand side of the linear equation. It could be the matrix of the column vectors and. It could be the matrix of the row vectors and. I don’t think I really know any more about them now than I did when I was confused. I no longer feel that way, but I think it’s just because I gave up. I spent years feeling like I didn’t know what a matrix truly was.
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