1 00:00:01,120 --> 00:00:03,640 Topic three is systems of linear equations. 2 00:00:08,540 --> 00:00:12,060 So what I'm building up to, is I want to be able to solve problems like this. 3 00:00:12,060 --> 00:00:15,650 So I have three equations and three unknowns. 4 00:00:15,650 --> 00:00:19,150 So the unknowns are x, y and z. 5 00:00:19,150 --> 00:00:24,290 And then each row here is one equation and I want to find a, I want to find 6 00:00:26,730 --> 00:00:31,250 x y and z that simultaneously satisfy all three of these equations. 7 00:00:35,740 --> 00:00:37,760 So there are two ways I can look at this 8 00:00:37,760 --> 00:00:40,230 if I, if I try and translate this into vectors. 9 00:00:40,230 --> 00:00:43,060 So I can look at something called the row picture. 10 00:00:43,060 --> 00:00:47,070 So that's trying to describe each one of these rows, and if I, 11 00:00:47,070 --> 00:00:50,180 if I do that, what I end up with is the dot product. 12 00:00:50,180 --> 00:00:54,960 So I'd have the dot product of the coefficient, so there's an invisible 13 00:00:54,960 --> 00:00:58,560 1 here, so if I just say X, I really mean 1 times X. 14 00:00:58,560 --> 00:00:59,819 That's where this 1 comes from. 15 00:01:01,000 --> 00:01:06,550 So, the first row is 1, 2, 3 dot X, Y, Z is equal to 6. 16 00:01:06,550 --> 00:01:11,240 Second row is 2, 5, 2 dot X, Y, Z is equal to 4 and so on. 17 00:01:13,180 --> 00:01:16,820 And it turns out I can also do this using a column picture. 18 00:01:20,180 --> 00:01:26,060 So I can also think of this as X times 1, 19 00:01:26,060 --> 00:01:31,880 2, 6, y as 2 5 minus 3 and z 20 00:01:31,880 --> 00:01:37,060 times 3, 2, 1, sounds just the same numbers are the coefficients here. 21 00:01:37,060 --> 00:01:40,400 So, I can think of this either as dot products, they give me the 22 00:01:40,400 --> 00:01:45,780 rows or I can think of this as a linear combination of the columns of 23 00:01:45,780 --> 00:01:47,490 my coefficients. 24 00:01:54,260 --> 00:01:58,990 So the problem this is trying to describe, let's see, so 25 00:01:58,990 --> 00:02:02,370 you can see there is sort of this dark point here. 26 00:02:02,370 --> 00:02:04,480 And each one of these equations in the 27 00:02:04,480 --> 00:02:07,740 row picture, it's describing one of these planes. 28 00:02:09,240 --> 00:02:13,110 And so it's, it's really difficult to try and put three planes onto one picture. 29 00:02:13,110 --> 00:02:16,820 Even, even this one, I failed miserably trying to 30 00:02:16,820 --> 00:02:19,280 do my own artwork, so I just copied this out 31 00:02:19,280 --> 00:02:20,710 of the book. 32 00:02:20,710 --> 00:02:24,780 And even there, I still can't really convince myself 33 00:02:24,780 --> 00:02:27,720 what direction this plane is supposed to be tilting. 34 00:02:29,926 --> 00:02:32,620 you know sometimes it goes from looking like this to looking like this to me. 35 00:02:34,500 --> 00:02:38,129 But the point of this is I'm supposed to have these two planes. 36 00:02:40,190 --> 00:02:44,380 And the space where they intersect is going to be this black line, here. 37 00:02:44,380 --> 00:02:47,250 So if I have one point that's kind of like this and one point 38 00:02:47,250 --> 00:02:51,790 that's like this, where they intersect is going to be described by a line. 39 00:02:51,790 --> 00:02:56,990 So the intersection of the first two planes will give me a line. 40 00:02:56,990 --> 00:03:00,430 So I've gone from having sort of two-dimensional objects. 41 00:03:01,830 --> 00:03:05,460 So these two planes, they're two-dimensional objects that exist 42 00:03:05,460 --> 00:03:06,380 in three dimensions. 43 00:03:06,380 --> 00:03:10,960 When I intersect them, I now have a one-dimensional object, so this line that 44 00:03:10,960 --> 00:03:16,400 exists in three dimensions, and then to solve my problem. 45 00:03:16,400 --> 00:03:19,340 So to finish solving my system of equations, I 46 00:03:19,340 --> 00:03:22,460 need to intersect that line with the third plane. 47 00:03:22,460 --> 00:03:27,260 And so, if I have a plane and one line, it's going to hit at just a single point. 48 00:03:27,260 --> 00:03:30,720 And that point is going to be the value of 49 00:03:30,720 --> 00:03:33,729 x, y, and z that solves my system of equations. 50 00:03:40,590 --> 00:03:44,370 So, I want to now try so, so that's sort of the row picture. 51 00:03:44,370 --> 00:03:46,010 So remember the rows were giving me 52 00:03:46,010 --> 00:03:48,830 equations of planes and those planes, then, were 53 00:03:48,830 --> 00:03:52,520 giving me this picture and that was giving me one approach for solving this problem. 54 00:03:53,600 --> 00:03:59,144 And now I'm also going to try and consider solving it using the, the column approach. 55 00:03:59,144 --> 00:04:05,620 So to do that I'm going to start by stacking rows, or binding columns. 56 00:04:05,620 --> 00:04:05,710 So 57 00:04:05,710 --> 00:04:09,040 I can either think of the, the rows as being the coefficients 58 00:04:09,040 --> 00:04:14,310 from each equation or the columns of being the coefficients for each variable. 59 00:04:14,310 --> 00:04:18,570 So if I stack the rows, I put the coefficients for the 60 00:04:18,570 --> 00:04:22,840 third equation on bottom, second equation in the middle, and the third 61 00:04:22,840 --> 00:04:27,470 equation on top, or I can take all of the x coor-, 62 00:04:27,470 --> 00:04:30,750 x coeffcients and put them in the first column, y coefficients go in 63 00:04:30,750 --> 00:04:33,970 the second column, z coefficients go in the third column. 64 00:04:33,970 --> 00:04:36,430 So, the, the nice thing about this is we get a duality. 65 00:04:36,430 --> 00:04:39,840 So it doesn't matter which approach I take, I'm going to get the same answer. 66 00:04:39,840 --> 00:04:45,490 So, quite often, like in, in statistics, it turns out that looking at the columns 67 00:04:45,490 --> 00:04:52,310 of a matrix makes calculating statistical properties of estimators very easy. 68 00:04:52,310 --> 00:04:56,110 And looking at the rows of the data in a matrix makes 69 00:04:56,110 --> 00:04:57,070 plotting the data. 70 00:04:57,070 --> 00:05:01,770 So, actually looking at it you know, with your eye, very easy. 71 00:05:01,770 --> 00:05:04,010 So, you can use the rows to make plots, 72 00:05:04,010 --> 00:05:06,940 and you can use the columns to do mathematics. 73 00:05:06,940 --> 00:05:10,200 But, because you're really doing the same thing, regardless of whether 74 00:05:10,200 --> 00:05:12,700 you try to use the row problem, or the column problem. 75 00:05:13,800 --> 00:05:17,890 both of those are going to be descriptions of the same actual problem. 76 00:05:21,410 --> 00:05:27,510 So if A is my coefficient matrix, then I can write my system of three 77 00:05:27,510 --> 00:05:34,110 equations and three unknowns like this. So I have the coefficient matrix. 78 00:05:34,110 --> 00:05:37,060 Times a, a vector of unknowns and that's going 79 00:05:37,060 --> 00:05:41,380 to equal a vector of the, the right hand sine. 80 00:05:41,380 --> 00:05:46,610 So the, you know these were equations like somethi- you know, 81 00:05:46,610 --> 00:05:51,920 a x plus b y plus c z equals d so these are these values d. 82 00:05:51,920 --> 00:05:54,780 They go on the right hand side here in a vector called b. 83 00:05:58,920 --> 00:06:03,990 And so you can recover pretty easily the row picture here, so I just have 1, 84 00:06:03,990 --> 00:06:08,220 2, 3 dot x, y ,z equals 6, that's going to give me the first equation. 85 00:06:08,220 --> 00:06:12,787 And I can repeat that for the second or the third row, or I can say x. 86 00:06:12,787 --> 00:06:19,280 Times 1, 2,6 Plus y times 2, 5, -3 Plus z times 3,2,1 87 00:06:19,280 --> 00:06:22,570 Equals 6, 4, 2 and that gives me the column picture of the problem. 88 00:06:29,130 --> 00:06:31,030 And so we can think of this as the left 89 00:06:31,030 --> 00:06:35,129 hand side multiplies a times the unknowns v to get v. 90 00:06:38,450 --> 00:06:41,070 And so what we want to move towards now is having some sort of 91 00:06:41,070 --> 00:06:44,290 multiplication rule for matrices and vectors that's 92 00:06:44,290 --> 00:06:45,940 going to make sense out of this. 93 00:06:45,940 --> 00:06:48,840 So the, the properties we want it to have are it has 94 00:06:48,840 --> 00:06:53,250 to give this linear combination for the column picture, and it needs to 95 00:06:54,670 --> 00:06:59,052 Give me the row picture, so I need the dot product here,uh you 96 00:06:59,052 --> 00:07:04,200 know that this row of this matrix times this column vector of unknowns, 97 00:07:04,200 --> 00:07:05,560 that needs to be a don't product. 98 00:07:08,850 --> 00:07:16,110 So I can represent this as Av being just row 1 dot v row 2 dot v or row 3 dot v. 99 00:07:19,050 --> 00:07:24,970 Or the column picture, a times v, that's just going to be, so x here was the first 100 00:07:24,970 --> 00:07:30,970 component of my vector v, y was the second component, and z was the third component. 101 00:07:33,320 --> 00:07:37,981 And so just some examples here, so if I have 1, so 1 102 00:07:37,981 --> 00:07:42,914 0 0.4 5 6 that's going to equal 103 00:07:42,914 --> 00:07:48,010 4. 1 0 0.4 5 6 104 00:07:48,010 --> 00:07:52,940 again is going to equal 4 and 1 0 0.4 5 6 is going to equal 4. 105 00:07:52,940 --> 00:07:58,520 So multiplying by this matrix A Just plucks 106 00:07:58,520 --> 00:08:02,120 this first value out and puts that in the answer. 107 00:08:04,360 --> 00:08:06,720 Another matrix- this matrix has a special name. 108 00:08:06,720 --> 00:08:08,140 It's called the identity matrix. 109 00:08:10,120 --> 00:08:12,800 If I have ones down the diagonal so the 110 00:08:12,800 --> 00:08:15,090 diagonal is where the row is equal to the column. 111 00:08:15,090 --> 00:08:16,970 So here I'm in the first row and first 112 00:08:16,970 --> 00:08:19,789 column, here I'm in the second row and second column. 113 00:08:20,850 --> 00:08:25,530 Here, I'm in the first row but the second column, so it's not on the diagonal. 114 00:08:25,530 --> 00:08:28,730 So if I have ones down the diagonal and zeroes everywhere else, 115 00:08:29,950 --> 00:08:33,390 then that's going to give me the same vector back. 116 00:08:33,390 --> 00:08:35,730 So I have one zero zero times four five 117 00:08:35,730 --> 00:08:38,620 six, that's going to just pluck out this four. 118 00:08:38,620 --> 00:08:39,510 Then 010 times 456. 119 00:08:39,510 --> 00:08:44,860 That's going to take the 5. Then 001, that's 120 00:08:44,860 --> 00:08:49,710 going to take this 6. 121 00:08:49,710 --> 00:08:55,220 So suppose I have this system 122 00:08:55,220 --> 00:08:58,140 of equations. So 3x minus Y equals 3. 123 00:08:58,140 --> 00:09:00,270 X plus Y equals 5. 124 00:09:00,270 --> 00:09:03,500 I can write that in the matrix form as a coefficient 125 00:09:03,500 --> 00:09:07,690 matrix times an unknown vector XY is equal to the vector 35. 126 00:09:07,690 --> 00:09:08,190 And 127 00:09:10,420 --> 00:09:12,210 I can draw two pictures of this now. 128 00:09:12,210 --> 00:09:14,005 So as long as these are, you know, 129 00:09:14,005 --> 00:09:17,060 two equations and 2 unknowns, it's easy to draw. 130 00:09:18,740 --> 00:09:23,430 So the row picture, this is probably how you want to think about this. 131 00:09:24,470 --> 00:09:27,923 So each one of these rows determines a line in the X-Y plane, 132 00:09:27,923 --> 00:09:32,260 and then the answer is just going to be where the two lines intersect. 133 00:09:32,260 --> 00:09:35,240 So I drew these two lines and I intersected at the point 2,3, 134 00:09:35,240 --> 00:09:41,540 and so you can see, if I put in 2,3 here, I get 6 minus 3 is equal to 3, and 135 00:09:41,540 --> 00:09:46,240 2 plus 3 is equal to 5 so that is actually the solution to this system of equations. 136 00:09:50,030 --> 00:09:54,170 I can also draw the column picture. So this one's a little bit trickier to 137 00:09:54,170 --> 00:10:01,110 see, but the columns are 3 1 so that gives me my first column. 138 00:10:01,110 --> 00:10:04,360 And so then this arrow here, labeled column one. 139 00:10:04,360 --> 00:10:07,795 I go over three and up one. So this is column one. 140 00:10:07,795 --> 00:10:15,290 Column 2 is -1, 1 so that's this vector here so -1 1 and 141 00:10:15,290 --> 00:10:16,450 it's labeled column two. 142 00:10:18,420 --> 00:10:23,020 And then I'm looking for a linear combination, so I'm looking for something 143 00:10:23,020 --> 00:10:28,170 times column 1 plus something times column 2 that's going to equal 3,5. 144 00:10:28,170 --> 00:10:33,580 And so you can visualize that, if I want to get,if I have to follow these two 145 00:10:33,580 --> 00:10:36,980 directions, so either this direction or this direction, 146 00:10:36,980 --> 00:10:38,770 and I want to get to the point 3,5. 147 00:10:38,770 --> 00:10:41,040 Then what 148 00:10:41,040 --> 00:10:47,040 I need to do is go 2 times column 1, and that gets me to this point. 149 00:10:47,040 --> 00:10:51,810 And then I'm going to add 3 times column 2, that gets me up to this point. 150 00:10:51,810 --> 00:10:58,780 So this point up here, which is 3,5, I can describe that as 2 times column 1. 151 00:10:58,780 --> 00:11:00,640 Plus 3 times column 2. 152 00:11:03,080 --> 00:11:06,950 So 2 times column 1. Plus 3 times column 2. 153 00:11:06,950 --> 00:11:13,040 So that gives me the vector 6, 2. Plus the vector -3, 3. 154 00:11:13,040 --> 00:11:15,994 So six minus 3 gives me 3. 155 00:11:15,994 --> 00:11:19,600 And 2, 3 gives me 5 so that gives me the, the same answer 156 00:11:19,600 --> 00:11:22,990 whether I try and do this using the row view or the column view. 157 00:11:24,900 --> 00:11:28,240 And then both of those are representations of this matrix 158 00:11:28,240 --> 00:11:29,110 form of the equation. 159 00:11:34,260 --> 00:11:38,720 So some properties of systems of equations so most of the time If you 160 00:11:38,720 --> 00:11:40,630 have the same number of equations and 161 00:11:40,630 --> 00:11:42,920 unknowns, there's going to be one solution.. 162 00:11:42,920 --> 00:11:47,100 So, that means, you know, you have a plane like this, a plane like this. 163 00:11:47,100 --> 00:11:49,800 That determines a line and then you maybe have a plane like 164 00:11:49,800 --> 00:11:53,040 this and wherever that line hits the third plane, that's your solution. 165 00:11:54,180 --> 00:11:56,230 But, of course, things can go wrong, so you 166 00:11:56,230 --> 00:12:00,310 could have two planes that are actually the same plane. 167 00:12:00,310 --> 00:12:02,500 So Imagine if I took the equation of a plane 168 00:12:02,500 --> 00:12:06,990 and just multiplied both sides by 2, every, every point 169 00:12:06,990 --> 00:12:11,310 would still be, the, the points that satisfy that are 170 00:12:11,310 --> 00:12:14,500 going to be the points in the same actual plane. 171 00:12:14,500 --> 00:12:16,870 Or I could have two planes that were parallel 172 00:12:16,870 --> 00:12:19,289 and so there would be no intersection between those. 173 00:12:23,040 --> 00:12:25,980 So, there may be no solution. 174 00:12:25,980 --> 00:12:29,700 So, if two lines are parallel and basically the, my interpretation 175 00:12:29,700 --> 00:12:33,300 of a system of equations is, where do these two lines intersect. 176 00:12:33,300 --> 00:12:36,100 If the two lines are parallel, we will get no solution. 177 00:12:37,810 --> 00:12:40,450 Or if you have two equations that 178 00:12:40,450 --> 00:12:42,970 look different but are actually describing the same 179 00:12:42,970 --> 00:12:45,770 line in the plane, then you have infinitely 180 00:12:45,770 --> 00:12:48,410 many solutions, so any point on that plane 181 00:12:48,410 --> 00:12:51,100 will satisfy the equation of both lines. 182 00:12:51,100 --> 00:12:54,960 Sorry, any point on that line satisfies the equation of each line.