Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
1
Academic year
2022
Sporkz
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57
p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:Arial;color:#000000;} .ft02{font-size:18px;font-family:ArialMT;color:#000000;} .ft03{font-size:18px;line-height:22px;font-family:Arial;color:#000000;} .ft04{font-size:18px;line-height:22px;font-family:ArialMT;color:#000000;} Identity matrix A matrix is a rectangular array of numbers or other data, usually arranged in rows andcolumns. A matrix can have any number of rows and columns. The most common type ofmatrix is the two-dimensional matrix, which has two dimensions (rows and columns).In mathematics and science, matrices are used to represent linear transformations, suchas rotation or reflection or for various other applications such as solving systems of linearequations. In computer graphics, they are used to manipulate coordinates and vectors.The dot product of two vectors is a scalar quantity that measures the extent to which thetwo vectors are pointing in the same direction. Let's say that we multiply the matrix I with a vector x1, x2 and x3. The first entry will be the dot product between 1, 0, 0 and x1, x2, x3. If you want this vector, it is written i hat, OK? So, if you take the dot product of i hat withitself, you will get the x component. The first component of the vector will be x1. That is equal to 1 times x1 plus 0, where 0 isthe zero vector. Similarly, if we do the dot product between two vectors, we get 0 plus x2plus 0. So this gives us x2. Here we get x3. OK? It makes sense. Also, when I put a matrixhere, I will get back the same matrix. Now, as you can see, the identity matrix in size n by n, so it's an n-by-n thing with 1s onthe diagonal and 0s everywhere else. OK? You can put a 1 at every diagonal position and 0 as well. Then if you multiply that bya vector, you will get the same vector back.
Identity Matrix in Mathematics and Science
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