Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
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52
What Are Matrices? A matrix is a rectangular array of numbers or other data. In mathematics, it is a rectangulararray of numbers whose elements are indexed by non-negative integers. Although theterm "matrix" can refer to diverse mathematical objects, its most familiar meaning is that ofan n-by-n table filled with elements whose ijth entry is denoted by aij. So maybe you have had a little exposure to matrices in high school, but if you haven't,here is just enough background so that you can follow along with the material. If you wantto know more about the life and times of matrices, then by all means take 1806 sometime. In life, many phenomena are related by linear formulas. Even when they are not andcannot be approximated by linear formulas, in some cases you can use linear formulas toapproximate them anyway. So often we find linear relations between variables. And for example, if we do a change ofcoordinate systems. So for example, say that we are in space. And we have a point. Itscoordinates in my initial coordinate system might be x1, x2, x3. But if I switch to a different coordinate system, I may be better able to solve my problembecause the new coordinates are more suited to other tasks I will perform in my solution.In these new coordinates, I will have other coordinate values—u1, u2 and u3. And then, if I choose the same origin, the relation between the old and the newcoordinates will be linear. I mean, especially if I choose different origins, otherwise theremight be constant terms which I won't insist on. So let me just give an example. Suppose, for example, that u1 equals twice x1 plus 3 times x2 plus 3 times x3. U2 mightequal 2 times x1 plus 4 times x2 plus 5 times x3. U3 might equal x1 plus x2 plus 2x3.Numbers come from nowhere, they are made up. OK, that's just an example of what might happen. So, if you want to express this kind oflinear relation in matrices, we can use matrix multiplication. So a matrix is just a table withnumbers in it and we can reformulate this in terms of matrix product. OK, so instead of writing this, I will write that for matrix 2, 3, 3; 2, 4, 5; 1, 1, 2 times thevector x1, x2, x3 is equal to u1, u2 and u3. Hopefully you see that there is the sameinformation content on both sides. I need to explain to you what this way of multiplyingtables of numbers means: we will take exactly these quantities. However, I would like to express this more symbolically. Let us call the first matrix A andthe second matrix X. Then we could say that A times X equals U. I need to explain what A, X, and U are, so that you can understand their entries in theformula. But this is convenient notation. The entries in a matrix product are obtained by dotting each row vector with itscorresponding column vector to obtain new row vectors. So, for example, we can computethe product A X by dotting the first row of A with the first column of X. We can perform the dot product between the rows of A and the columns of X. A is a 3x3matrix, meaning that there are three rows and three columns in A. We can think of X as a3-element column vector, or a 1-by-3 matrix. It has three rows and only one column.
Now, what can we do? As said, we would calculate the dot product between a row of A--2,3, 3 and a column of X--x1, x2, x3. The dot-product of two vectors, x1 and x2, results in 2x1 plus 3x2 and 3x3. Since this iswhat we want to set u1 equal to, we are done with this step. Let's do the second one. I take the second row of A—2, 4, 5 and multiply it by x1x2x3. I get2x1 plus 4x2 plus 5x3, which is this thing. And same with the third one. 1x1 plus 1x2 plus2x3. OK, so that was matrix multiplication.
Matrices: Understanding the Basics and Applications
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