WEBVTT 00:00:00.440 --> 00:00:05.360 Linear algebra is heavily used in programming, especially for graphics. 00:00:05.840 --> 00:00:08.470 To implement 3D scenes with scaling, 00:00:08.470 --> 00:00:12.790 rotations, and transformations, we need to operate on data 00:00:12.790 --> 00:00:16.550 presented with mathematical vectors and matrices. 00:00:16.940 --> 00:00:19.020 And, if you ever saw a matrix, 00:00:19.030 --> 00:00:22.080 you know that it kind of looks like a table of numbers with 00:00:22.080 --> 00:00:24.760 a specific amount of rows and columns. 00:00:25.140 --> 00:00:28.430 So, let's say that I want to store a matrix in memory, 00:00:28.440 --> 00:00:30.360 that looks something like this. 00:00:30.740 --> 00:00:34.060 It has two rows and three columns. 00:00:34.440 --> 00:00:39.750 Remember that to the computer, everything is just a really long stream of bytes. 00:00:40.140 --> 00:00:43.460 The concept of rows and columns is not something that 00:00:43.460 --> 00:00:45.420 the computer really cares about. 00:00:45.430 --> 00:00:48.220 If we wanted to store this matrix in memory, 00:00:48.230 --> 00:00:51.940 each element would be stored in a one‑dimensional sequence, 00:00:51.950 --> 00:00:56.610 just like a simple array. I created an array of six integers to 00:00:56.610 --> 00:00:58.850 store the numbers from the matrix. 00:00:59.340 --> 00:01:02.570 This doesn't really look like a table with rows and columns, 00:01:02.570 --> 00:01:06.260 but we can make our own abstraction to pretend that it does. 00:01:06.840 --> 00:01:08.530 Let's create four variables. 00:01:08.540 --> 00:01:12.350 The first two will define the number of rows and the number of 00:01:12.350 --> 00:01:16.350 columns, and the last two will store the indexes of a row and 00:01:16.350 --> 00:01:18.960 the column that we want to access. 00:01:19.440 --> 00:01:24.350 Our table has two rows and each row has three columns, so 00:01:24.350 --> 00:01:28.420 six numbers in total. The required row and column position 00:01:28.420 --> 00:01:30.360 will be taken from the terminal. 00:01:30.740 --> 00:01:32.630 I want to start counting from 0, 00:01:32.910 --> 00:01:36.320 which means that the first row will have an index of 0 and 00:01:36.320 --> 00:01:38.710 the second row will have an index of 1, 00:01:38.990 --> 00:01:42.790 just like a regular array. Each column will also have 00:01:42.790 --> 00:01:44.860 a special index for each number. 00:01:45.240 --> 00:01:46.260 So for example, 00:01:46.260 --> 00:01:49.230 if I want to get this number, which is placed in the 00:01:49.230 --> 00:01:51.430 second column of the second row, 00:01:51.440 --> 00:01:53.810 I would say that I want to get an integer at the 00:01:53.810 --> 00:01:56.630 position of the row 1 and column 1. 00:01:57.440 --> 00:02:00.620 I need a mathematical formula that will provide me with 00:02:00.620 --> 00:02:03.260 this abstraction of two dimensions. 00:02:04.640 --> 00:02:07.460 And, this is what that formula looks like. 00:02:07.840 --> 00:02:11.980 I want to get an array element at the position of a requested row, 00:02:11.990 --> 00:02:16.560 multiplied by the number of columns, plus the requested column. 00:02:17.740 --> 00:02:20.560 Let's compile this program to see if this works. 00:02:20.940 --> 00:02:25.900 I want to get a second number from the second row, so the position that I need 00:02:25.910 --> 00:02:32.560 is 1, 1, and it seems that it works, the number 5 is correct. 00:02:33.740 --> 00:02:35.150 Let's try one more. 00:02:35.840 --> 00:02:41.870 I want to get this number 3 and its position is 0 and 2, and if I 00:02:41.870 --> 00:02:44.950 try that position, I get the correct number again. 00:02:45.340 --> 00:02:49.810 So, with a little bit of math and imagination, we were able to use this 00:02:49.810 --> 00:02:53.650 one‑dimensional array as a matrix with two dimensions. 00:02:54.040 --> 00:02:57.600 Let's use this same formula to print this array to the terminal 00:02:57.610 --> 00:03:03.140 in a form of a table. To do this, I will use two for loops, one 00:03:03.140 --> 00:03:05.250 will be nested inside of the other. 00:03:06.240 --> 00:03:09.230 The first loop will go through each row, and the 00:03:09.230 --> 00:03:12.150 second loop will go over each column. 00:03:12.540 --> 00:03:17.250 Each iteration of this outer loop will complete the whole inner loop. 00:03:17.640 --> 00:03:21.870 So first, the row will be 0 and the column will also be 0. 00:03:22.540 --> 00:03:26.960 Then the column will be incremented to 1, but the row will still 00:03:26.960 --> 00:03:31.610 remain 0. After that, the column will increment once more to 2, 00:03:31.610 --> 00:03:33.850 and the row will still remain 0. 00:03:34.540 --> 00:03:38.470 So, we are basically looping through all of the numbers in the table, 00:03:38.480 --> 00:03:42.650 row by row. Once all of the columns in a row are completed, 00:03:42.660 --> 00:03:46.500 the next iteration of the outer loop begins, and this goes 00:03:46.500 --> 00:03:49.360 on until we get through all of the rows. 00:03:50.840 --> 00:03:54.760 So the actual elements from the array need to be accessed 00:03:54.760 --> 00:03:58.600 here, inside of the nested for loop. I will use the same 00:03:58.600 --> 00:04:00.850 formula to access these elements. 00:04:01.540 --> 00:04:05.490 Let's also add this vertical line character after each number 00:04:05.490 --> 00:04:08.050 to make the output look more like a table. 00:04:08.940 --> 00:04:12.030 I will also add the same character before each row, 00:04:12.040 --> 00:04:16.260 and after each row is finished, I want to print on a new line. 00:04:16.640 --> 00:04:19.820 This is a really useful concept, because you can define 00:04:19.820 --> 00:04:23.930 special behavior for each row in the outer loop and what to do 00:04:23.930 --> 00:04:26.360 with each column in the inner loop. 00:04:27.540 --> 00:04:31.560 I will turn this into a comment, because I just want to see the table. 00:04:32.640 --> 00:04:34.860 Let's compile and see what happens. 00:04:36.340 --> 00:04:39.360 This kind of looks like a table, right? 00:04:39.740 --> 00:04:43.630 It looks exactly like the matrix we had in mind, and now we have 00:04:43.630 --> 00:04:46.660 a way to store it in memory and work with it. 00:04:47.040 --> 00:04:47.660 However, 00:04:47.660 --> 00:04:51.470 using the array with this mathematical formula seems a little bit 00:04:51.480 --> 00:04:56.700 unnatural and unintuitive, at least for me. There is a reason why we 00:04:56.700 --> 00:05:02.180 don't program by typing out a sequence of 1s and 0s, humans need better 00:05:02.180 --> 00:05:04.860 abstraction mechanisms to be more productive. 00:05:05.540 --> 00:05:10.760 Fortunately, C++ offers us a better way to manage multidimensional arrays. 00:05:11.840 --> 00:05:14.130 The syntax is straightforward, 00:05:14.140 --> 00:05:18.160 you just use a separate square bracket to define each dimension. 00:05:18.540 --> 00:05:23.550 So in this case, I want to create a 2d array with two rows and three columns. 00:05:24.040 --> 00:05:28.030 The initialization list can stay, because the memory which contains 00:05:28.030 --> 00:05:31.940 these elements will look exactly the same, it will still store all 00:05:31.940 --> 00:05:33.960 of these integers next to each other. 00:05:34.340 --> 00:05:37.030 The big difference is, of course, the way in which we 00:05:37.030 --> 00:05:39.370 can manage this array. However, 00:05:39.370 --> 00:05:42.740 to make our intentions more obvious, you can wrap every 00:05:42.740 --> 00:05:45.210 dimension in its own curly braces, 00:05:45.220 --> 00:05:48.650 and I will even put each one of them on a new line. 00:05:49.540 --> 00:05:53.460 This already looks like a table, and now, 00:05:53.460 --> 00:05:55.600 instead of utilizing this formula, 00:05:55.610 --> 00:05:58.470 I can use two square brackets to specify the 00:05:58.470 --> 00:06:01.260 required position of each dimension. 00:06:02.040 --> 00:06:05.460 As you can see, this approach is much more intuitive. 00:06:06.540 --> 00:06:07.130 Okay, 00:06:07.140 --> 00:06:10.880 now I can compile this to see if it still works, and it 00:06:10.880 --> 00:06:15.760 does. The output still looks the same, but the code is much cleaner. 00:06:16.140 --> 00:06:18.760 If you wanted to have a three‑dimensional array, 00:06:18.770 --> 00:06:22.720 you would just add another square brackets with the third dimension and then 00:06:22.730 --> 00:06:27.480 access the elements in the same way. You should prefer using this syntax 00:06:27.490 --> 00:06:29.960 instead of the implementation with a special formula, 00:06:29.970 --> 00:06:34.950 because the code looks cleaner and the performance is exactly the same. 00:06:35.340 --> 00:06:38.780 But there is a reason why I started this lesson with that formula, 00:06:38.790 --> 00:06:41.120 and we will talk about it in the next lesson, 00:06:41.120 --> 00:06:45.750 where we will learn how to allocate multidimensional arrays on the heap.