1 00:00:00,000 --> 00:00:00,610 2 00:00:00,610 --> 00:00:03,810 I have here three linear equations of four unknowns. 3 00:00:03,810 --> 00:00:07,050 And like the first video, where I talked about reduced 4 00:00:07,049 --> 00:00:10,689 row echelon form, and solving systems of linear equations 5 00:00:10,689 --> 00:00:13,509 using augmented matrices, at least my gut feeling says, 6 00:00:13,509 --> 00:00:16,719 look, I have fewer equations than variables, so I probably 7 00:00:16,719 --> 00:00:18,469 won't be able to constrain this enough. 8 00:00:18,469 --> 00:00:20,369 Or maybe I'll have an infinite number of solutions. 9 00:00:20,370 --> 00:00:21,690 But let's see if I'm right. 10 00:00:21,690 --> 00:00:24,720 So let's construct the augmented matrix for this 11 00:00:24,719 --> 00:00:27,309 system of equations. 12 00:00:27,309 --> 00:00:33,189 My coefficients on the x1 terms are 1, 1, and 2. 13 00:00:33,189 --> 00:00:35,229 Coefficients on the x2 are 2, 2, and 4. 14 00:00:35,229 --> 00:00:38,669 15 00:00:38,670 --> 00:00:43,679 Coefficients on x3 are 1, 2, and 0. 16 00:00:43,679 --> 00:00:46,189 There's of course no x3 term, so we can view it as a 0 17 00:00:46,189 --> 00:00:47,059 coefficient. 18 00:00:47,060 --> 00:00:49,750 Coefficients on the x4 are 1, minus 1, and 6. 19 00:00:49,750 --> 00:00:53,939 20 00:00:53,939 --> 00:00:56,079 And then on the right-hand side of the equals sign, I 21 00:00:56,079 --> 00:01:00,670 have 8, 12, and 4. 22 00:01:00,670 --> 00:01:04,359 There's my augmented matrix, now let's put this guy into 23 00:01:04,359 --> 00:01:08,430 reduced row echelon form. 24 00:01:08,430 --> 00:01:10,640 The first thing I want to do is, I want to zero out these 25 00:01:10,640 --> 00:01:12,500 two rows right here. 26 00:01:12,500 --> 00:01:14,909 So what can we do? 27 00:01:14,909 --> 00:01:18,549 I'm going to keep my first row the same for now, so it's 1, 28 00:01:18,549 --> 00:01:23,450 2, 1, 1, 8. 29 00:01:23,450 --> 00:01:28,079 That line essentially represents my equals sign. 30 00:01:28,079 --> 00:01:30,670 What I can do is, let me subtract-- let me replace the 31 00:01:30,670 --> 00:01:35,170 second row with the second row minus the first row. 32 00:01:35,170 --> 00:01:45,734 So 1 minus 1 is 0, 2 minus 2 is 0, 2 minus 1 is 1, 1-- 33 00:01:45,734 --> 00:01:54,530 negative 1 minus 1 is minus 2, and then 12 minus 8 is 4. 34 00:01:54,530 --> 00:01:58,140 There you go, that looks good so far, it looks like column, 35 00:01:58,140 --> 00:02:00,680 or x2 which is represented by column two, looks like it 36 00:02:00,680 --> 00:02:04,590 might be a free variable, but we're not 100% sure yet. 37 00:02:04,590 --> 00:02:07,240 Let's do all of our rows. 38 00:02:07,239 --> 00:02:11,909 So let's take-- to get rid of this guy right here, let's 39 00:02:11,909 --> 00:02:14,620 replace our third equation with the third equation minus 40 00:02:14,620 --> 00:02:17,719 two times our first equation. 41 00:02:17,719 --> 00:02:26,949 So we get 2 minus 2 times 1 is 0, 4 minus 2 times 2, well 42 00:02:26,949 --> 00:02:35,799 that's 0, 0 minus 2 times 1, that's minus 2. 43 00:02:35,800 --> 00:02:40,330 6 minus 2 times 1, well that's 4, right? 44 00:02:40,330 --> 00:02:41,600 6 minus 2. 45 00:02:41,599 --> 00:02:47,379 And then 4 minus 2 times 8, is minus 16, 4 46 00:02:47,379 --> 00:02:52,509 minus 16 is minus 12. 47 00:02:52,509 --> 00:02:53,569 Now what can we do? 48 00:02:53,569 --> 00:02:56,739 Well, let's see if we can get rid of this minus 49 00:02:56,740 --> 00:02:58,100 2 term right there. 50 00:02:58,099 --> 00:03:03,639 So let me rewrite my augmented matrix. 51 00:03:03,639 --> 00:03:07,509 I'm going to keep row two the same this time, so I get a 0, 52 00:03:07,509 --> 00:03:12,389 0, 1, minus 2, and essentially my equals sign, or the 53 00:03:12,389 --> 00:03:15,069 augmented part of the matrix. 54 00:03:15,069 --> 00:03:17,699 And now let's see what I can do. 55 00:03:17,699 --> 00:03:19,563 Let me get rid of this 0 up here, because I want to get 56 00:03:19,563 --> 00:03:21,090 into reduced row echelon form. 57 00:03:21,090 --> 00:03:24,170 So any of my pivot entries, which are always going to have 58 00:03:24,169 --> 00:03:26,799 the coefficient 1, or the entry 1, it should be the only 59 00:03:26,800 --> 00:03:28,560 non-zero term in my row. 60 00:03:28,560 --> 00:03:30,909 How do I get rid of this one here? 61 00:03:30,909 --> 00:03:35,719 Well I can subtract-- I can replace row one with row 1 62 00:03:35,719 --> 00:03:38,000 minus row 2. 63 00:03:38,000 --> 00:03:48,379 So 1 minus 0 is 1, 2 minus 0 is 2, 1 minus 1 is 0, 1 minus 64 00:03:48,379 --> 00:03:52,519 minus 2, that's 1 plus 2, which is 3. 65 00:03:52,520 --> 00:03:56,750 And then 8 minus 4 is 4. 66 00:03:56,750 --> 00:03:59,199 And now how can I get rid of this guy? 67 00:03:59,199 --> 00:04:06,909 Well let me replace row 3 with row 3 plus 2 times row 1. 68 00:04:06,909 --> 00:04:09,780 Sorry, with row 3 plus 2 times row 2. 69 00:04:09,780 --> 00:04:10,050 Right? 70 00:04:10,050 --> 00:04:12,600 Because then you'd have minus 2, plus 2 times this, and 71 00:04:12,599 --> 00:04:13,409 they'd cancel out. 72 00:04:13,409 --> 00:04:15,069 So let's see the zeros. 73 00:04:15,069 --> 00:04:18,110 0 plus 2 times 0, that's 0. 74 00:04:18,110 --> 00:04:26,120 0 plus 2 times 0, that's 0, minus 2 plus 2 times 1 is 0. 75 00:04:26,120 --> 00:04:35,220 4 plus 2 times minus 2, that's 4 minus 4, that's 0. 76 00:04:35,220 --> 00:04:40,410 And then I have minus 12, plus 2 times 4. 77 00:04:40,410 --> 00:04:44,175 That's minus 12 plus 8, that's minus 4. 78 00:04:44,175 --> 00:04:47,040 79 00:04:47,040 --> 00:04:49,680 Now, this is interesting right now-- this is interesting. 80 00:04:49,680 --> 00:04:53,170 I have essentially put this in reduced row echelon form. 81 00:04:53,170 --> 00:04:56,720 I have two pivot entries, that's a pivot entry right 82 00:04:56,720 --> 00:04:59,230 there, and that's a pivot entry right there. 83 00:04:59,230 --> 00:05:03,230 They're the only non-zero term in their respective columns. 84 00:05:03,230 --> 00:05:05,420 And this is just kind of a style issue, but this pivot 85 00:05:05,420 --> 00:05:08,009 entry is in a lower row than that one. 86 00:05:08,009 --> 00:05:12,930 So it's in a column to the right of this one right there. 87 00:05:12,930 --> 00:05:16,860 And I just inspected, this looks like a-- this column two 88 00:05:16,860 --> 00:05:19,120 looks kind of like a free variable-- there's no pivot 89 00:05:19,120 --> 00:05:20,410 entry there, no pivot entry there. 90 00:05:20,410 --> 00:05:22,820 But let's see, let's map this back to 91 00:05:22,819 --> 00:05:24,860 our system of equations. 92 00:05:24,860 --> 00:05:26,990 These are just numbers to me and I just kind of 93 00:05:26,990 --> 00:05:29,600 mechanically, almost like a computer, put this in reduced 94 00:05:29,600 --> 00:05:31,010 row echelon form. 95 00:05:31,009 --> 00:05:32,529 Actually, almost exactly like a computer. 96 00:05:32,529 --> 00:05:35,000 But let me put it back to my system of linear equations, to 97 00:05:35,000 --> 00:05:37,129 see what our result is. 98 00:05:37,129 --> 00:05:41,610 So we get 1 times x1, let me write it in yellow. 99 00:05:41,610 --> 00:05:50,930 So I get 1 times x1, plus 2 times x2, plus 0 times x3, 100 00:05:50,930 --> 00:05:55,079 plus 3 times x4 is equal to 4. 101 00:05:55,079 --> 00:05:57,099 Obviously I could ignore this term right there, I didn't 102 00:05:57,100 --> 00:05:58,930 even have to write it. 103 00:05:58,930 --> 00:05:59,220 Actually. 104 00:05:59,220 --> 00:05:59,980 I'm not going to write that. 105 00:05:59,980 --> 00:06:05,210 Then I get 0 times x1, plus 0 times x2, plus 1 times x3, so 106 00:06:05,209 --> 00:06:05,859 I can just write that. 107 00:06:05,860 --> 00:06:06,850 I'll just write the one. 108 00:06:06,850 --> 00:06:12,650 1 times x3, minus 2 times x4, is equal to 4. 109 00:06:12,649 --> 00:06:14,579 And then this last term, what do I get? 110 00:06:14,579 --> 00:06:19,029 I get 0 x1, plus 0 x2 plus 0 x3 plus 0 x4, well, all of 111 00:06:19,029 --> 00:06:21,079 that's equal to 0, and I've got to write something on the 112 00:06:21,079 --> 00:06:22,409 left-hand side. 113 00:06:22,410 --> 00:06:25,240 So let me just write a 0, and that's got to be 114 00:06:25,240 --> 00:06:27,800 equal to minus 4. 115 00:06:27,800 --> 00:06:31,069 Well this doesn't make any sense whatsoever. 116 00:06:31,069 --> 00:06:33,060 0 equals minus 4. 117 00:06:33,060 --> 00:06:36,160 This is this is a nonsensical constraint, this is 118 00:06:36,160 --> 00:06:37,230 impossible. 119 00:06:37,230 --> 00:06:40,430 0 can never equal minus 4. 120 00:06:40,430 --> 00:06:42,653 This is impossible. 121 00:06:42,653 --> 00:06:46,050 122 00:06:46,050 --> 00:06:49,420 Which means that it is essentially impossible to find 123 00:06:49,420 --> 00:06:53,550 an intersection of these three systems of equations, or a 124 00:06:53,550 --> 00:06:56,030 solution set that satisfies all of them. 125 00:06:56,029 --> 00:06:58,219 When we looked at this initially, at the beginning of 126 00:06:58,220 --> 00:07:00,650 the of the video, we said there are only three 127 00:07:00,649 --> 00:07:02,169 equations, we have four unknowns, maybe there's going 128 00:07:02,170 --> 00:07:03,980 to be an infinite set of solutions. 129 00:07:03,980 --> 00:07:07,330 But turns out that these three-- I guess you can call 130 00:07:07,329 --> 00:07:11,319 them these three surfaces-- don't intersect in r4. 131 00:07:11,319 --> 00:07:11,560 Right? 132 00:07:11,560 --> 00:07:14,970 These are all four dimensional, we're dealing in 133 00:07:14,970 --> 00:07:19,080 r4 right here, because we have-- I guess each vector has 134 00:07:19,079 --> 00:07:22,209 four components, or we have four variables, is the way you 135 00:07:22,209 --> 00:07:23,129 could think about it. 136 00:07:23,129 --> 00:07:26,100 And it's always hard to visualize things in r4. 137 00:07:26,100 --> 00:07:29,350 But if we were doing things in r3, we can imagine the 138 00:07:29,350 --> 00:07:34,560 situation where, let's say we had two planes in r3. 139 00:07:34,560 --> 00:07:36,680 So that's one plane right there, and then I had another 140 00:07:36,680 --> 00:07:39,930 completely parallel plane to that one. 141 00:07:39,930 --> 00:07:42,110 So I had another completely parallel plane 142 00:07:42,110 --> 00:07:43,490 to that first one. 143 00:07:43,490 --> 00:07:47,319 Even though these would be two planes in r3, so let me give 144 00:07:47,319 --> 00:07:47,909 an example. 145 00:07:47,910 --> 00:07:51,640 So let's say that this first plane was represented by the 146 00:07:51,639 --> 00:08:02,680 equation 3x plus 6y plus 9z is equal to 5, and the second 147 00:08:02,680 --> 00:08:08,400 plane was represented by the equation 3x plus 6y plus 9z is 148 00:08:08,399 --> 00:08:11,169 equal to 2. 149 00:08:11,170 --> 00:08:15,629 These two planes in r3-- this is the case of r3, so this is 150 00:08:15,629 --> 00:08:17,540 r3 right here. 151 00:08:17,540 --> 00:08:19,810 These two planes, clearly they'll never intersect. 152 00:08:19,810 --> 00:08:23,519 Because obviously this one has the same coefficients adding 153 00:08:23,519 --> 00:08:26,430 up to 5, this one has the same coefficient adding up to 2. 154 00:08:26,430 --> 00:08:28,689 And when, if we just looked at this initially, if it wasn't 155 00:08:28,689 --> 00:08:31,089 so obvious, we would have said, we have only two 156 00:08:31,089 --> 00:08:33,529 equations with three unknowns, maybe this has an infinite set 157 00:08:33,529 --> 00:08:34,230 of solutions. 158 00:08:34,230 --> 00:08:36,658 But it won't be the case, because you can actually just 159 00:08:36,658 --> 00:08:39,798 subtract this equation, from the bottom equation, from the 160 00:08:39,798 --> 00:08:40,459 top equation. 161 00:08:40,460 --> 00:08:41,570 And what do you get? 162 00:08:41,570 --> 00:08:44,190 You would get a very familiar-- so if you just 163 00:08:44,190 --> 00:08:46,220 subtract the bottom equation from the top equation, and you 164 00:08:46,220 --> 00:08:50,182 get 3x minus 3x, 6y minus 6y, 9z minus 9z-- actually, let me 165 00:08:50,182 --> 00:08:51,500 do it right here. 166 00:08:51,500 --> 00:08:55,299 So for that minus that, you get zero is equal to 5 minus 167 00:08:55,299 --> 00:08:56,699 2, which is 3. 168 00:08:56,700 --> 00:08:59,200 Which is a very similar result that we got up there. 169 00:08:59,200 --> 00:09:03,110 So when you have two parallel planes, in this case in r3, or 170 00:09:03,110 --> 00:09:07,149 really to any kind of two parallel equations, or a set 171 00:09:07,149 --> 00:09:08,620 of parallel equations, they won't intersect. 172 00:09:08,620 --> 00:09:12,289 And you're going to get, when you put it in reduced row 173 00:09:12,289 --> 00:09:14,490 echelon form, or you just do basic elimination, or you 174 00:09:14,490 --> 00:09:17,250 solve the systems, you're going to get a statement that 175 00:09:17,250 --> 00:09:19,720 zero is equal to something, and that means that there are 176 00:09:19,720 --> 00:09:20,970 no solutions. 177 00:09:20,970 --> 00:09:23,990 178 00:09:23,990 --> 00:09:26,029 So the general take-away, if you have zero equals 179 00:09:26,029 --> 00:09:27,839 something, no solutions. 180 00:09:27,840 --> 00:09:31,769 If you have the same number of pivot variables, the same 181 00:09:31,769 --> 00:09:36,210 number of pivot entries as you do columns, so if you get the 182 00:09:36,210 --> 00:09:39,040 situations-- let me write this down, this is good to know. 183 00:09:39,039 --> 00:09:41,599 if you have zero is equal to anything, then 184 00:09:41,600 --> 00:09:44,759 that means no solution. 185 00:09:44,759 --> 00:09:46,710 If you're dealing with r3, then you probably have 186 00:09:46,710 --> 00:09:50,680 parallel planes, in r2 you're dealing with parallel lines. 187 00:09:50,679 --> 00:09:55,929 If you have the situation where you have the same number 188 00:09:55,929 --> 00:10:02,959 of pivot entries as columns, so it's just 1, 1, 1, 1, this 189 00:10:02,960 --> 00:10:05,350 is the case of r4 right there. 190 00:10:05,350 --> 00:10:07,389 I think you get the idea. 191 00:10:07,389 --> 00:10:10,439 That equals a, b, c, d. 192 00:10:10,440 --> 00:10:11,690 Then you have a unique solution. 193 00:10:11,690 --> 00:10:15,730 194 00:10:15,730 --> 00:10:19,009 Now if, you have any free variables-- so free variables 195 00:10:19,009 --> 00:10:23,189 look like this, so let's say we have 1, 0, 1, 0, and then I 196 00:10:23,190 --> 00:10:30,010 have the entry 1, 1, let me be careful. 197 00:10:30,009 --> 00:10:33,319 0, let me do it like this. 198 00:10:33,320 --> 00:10:41,040 1, 0, 0, and then I have the entry 1, 2, and then I have a 199 00:10:41,039 --> 00:10:43,519 bunch of zeroes over here. 200 00:10:43,519 --> 00:10:45,370 And then this has to equal zero-- remember, if this was a 201 00:10:45,370 --> 00:10:47,759 bunch of zeroes equaling some variable, then I would have no 202 00:10:47,759 --> 00:10:50,330 solution, or equalling some constant, let's say this is 203 00:10:50,330 --> 00:10:52,820 equal to 5, this is equal to 2. 204 00:10:52,820 --> 00:10:55,120 If this is our reduced row echelon form that we 205 00:10:55,120 --> 00:10:58,860 eventually get to, then we have a few free variables. 206 00:10:58,860 --> 00:11:02,750 This is a free, or I guess we could call this column a free 207 00:11:02,750 --> 00:11:04,730 column, to some degree this one would be too. 208 00:11:04,730 --> 00:11:07,389 Because it has no pivot entries. 209 00:11:07,389 --> 00:11:09,279 These are the pivot entries. 210 00:11:09,279 --> 00:11:13,419 So this is variable x2 and that's variable x4. 211 00:11:13,419 --> 00:11:14,909 Then these would be free, we can set 212 00:11:14,909 --> 00:11:16,240 them equal to anything. 213 00:11:16,240 --> 00:11:19,169 So then here we have unlimited solutions, 214 00:11:19,169 --> 00:11:21,669 or no unique solutions. 215 00:11:21,669 --> 00:11:23,509 And that was actually the first example we saw. 216 00:11:23,509 --> 00:11:25,539 And these are really the three cases that you're going to see 217 00:11:25,539 --> 00:11:27,879 every time, and it's good to get familiar with them so 218 00:11:27,879 --> 00:11:29,570 you're never going to get stumped up when you have 219 00:11:29,570 --> 00:11:32,930 something like 0 equals minus 4, or 0 equals 3. 220 00:11:32,929 --> 00:11:35,439 Or if you have just a bunch of zeros and a bunch of rows. 221 00:11:35,440 --> 00:11:36,870 I want to make that very clear. 222 00:11:36,870 --> 00:11:40,549 Sometimes, you see a bunch of zeroes here, on the left-hand 223 00:11:40,549 --> 00:11:44,359 side of the augmented divide, and you might say, oh maybe I 224 00:11:44,360 --> 00:11:45,769 have no unique solutions, I have an 225 00:11:45,769 --> 00:11:46,730 infinite number of solutions. 226 00:11:46,730 --> 00:11:50,379 But you have to look at this entry right here. 227 00:11:50,379 --> 00:11:52,879 Only if this whole thing is zero and you have free 228 00:11:52,879 --> 00:11:56,210 variables, then you have an infinite number of solutions. 229 00:11:56,210 --> 00:11:59,050 If you have a statement like, 0 is equal to a, if this is 230 00:11:59,049 --> 00:12:01,379 equal to 7 right here, then all of the sudden you would 231 00:12:01,379 --> 00:12:03,000 have no solution to this. 232 00:12:03,000 --> 00:12:05,389 That you're dealing with parallel surfaces. 233 00:12:05,389 --> 00:12:07,850 Anyway, hopefully you found that helpful.