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I have here three equations of four unknowns.

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You can already guess, or you already know, that if you have

4
00:00:07,530 --> 00:00:09,539
more unknowns than equations, you are probably not

5
00:00:09,539 --> 00:00:11,339
constraining it enough.

6
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You actually are going to have an

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infinite number of solutions.

8
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Those infinite number of solutions could still be

9
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constrained.

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Let's say we're in four dimensions, in this case,

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because we have four variables.

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Maybe we were constrained into a plane in four dimensions, or

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if we were in three dimensions, maybe we're

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constrained to a line.

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A line is an infinite number of solutions, but it's a more

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constrained set.

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Let's solve this set of linear equations.

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We've done this by elimination in the past. What I want to do

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is I want to introduce the idea of matrices.

20
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The matrices are really just arrays of numbers that are

21
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shorthand for this system of equations.

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Let me create a matrix here.

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I could just create a coefficient matrix, where the

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coefficient matrix would just be, let me write it neatly,

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the coefficient matrix would just be the coefficients on

26
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the left hand side of these linear equations.

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The coefficient there is 1.

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The coefficient there is 1.

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The coefficient there is 2.

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You have 2, 2, 4.

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2, 2, 4.

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1, 2, 0.

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1, 2, there is no coefficient the x3 term here, because

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there is no x3 term there.

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We'll say the coefficient on the x3 term there is 0.

36
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And then we have 1, minus 1, and 6.

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Now if I just did this right there, that would be the

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coefficient matrix for this system of

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equations right there.

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What I want to do is I want to augment it, I want to augment

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it with what these equations need to be equal to.

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Let me augment it.

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What I am going to do is I'm going to just draw a little

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line here, and write the 7, the 12, and the 4.

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I think you can see that this is just another

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way of writing this.

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And just by the position, we know that these are the

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coefficients on the x1 terms. We know that these are the

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coefficients on the x2 terms. And what this does, it really

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just saves us from having to write x1 and x2 every time.

51
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We can essentially do the same operations on this that we

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otherwise would have done on that.

53
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What we can do is, we can replace any equation with that

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equation times some scalar multiple,

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plus another equation.

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We can divide an equation, or multiply an

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equation by a scalar.

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We can subtract them from each other.

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We can swap them.

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Let's do that in an attempt to solve this equation.

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The first thing I want to do, just like I've done in the

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past, I want to get this equation into the form of,

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where if I can, I have a 1.

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My leading coefficient in any of my rows is a 1.

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And that every other entry in that column is a 0.

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In the past, I made sure that every other entry

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below it is a 0.

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That's what I was doing in some of the previous videos,

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when we tried to figure out of things were linearly

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independent, or not.

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Now I'm going to make sure that if there is a 1, if there

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is a leading 1 in any of my rows, that everything else in

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that column is a 0.

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That form I'm doing is called reduced row echelon form.

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Let me write that.

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Reduced row echelon form.

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If we call this augmented matrix, matrix A, then I want

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to get it into the reduced row echelon form of matrix A.

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And matrices, the convention is, just like vectors, you

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make them nice and bold, but use capital letters, instead

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of lowercase letters.

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We'll talk more about how matrices relate to vectors in

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the future.

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Let's just solve this system of equations.

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The first thing I want to do is, in an ideal world I would

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get all of these guys right here to be 0.

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Let me replace this guy with that guy, with the first entry

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minus the second entry.

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Let me do that.

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The first row isn't going to change.

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It's going to be 1, 2, 1, 1.

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And then I get a 7 right there.

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That's my first row.

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Now the second row, I'm going to replace it with the first

97
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row minus the second row.

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So what do I get.

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1 minus 1 is 0.

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2 minus 2 is 0.

101
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1 minus 2 is minus 1.

102
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And then 1 minus minus 1 is 2.

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That's 1 plus 1.

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And then 7 minus 12 is minus 5.

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Now I want to get rid of this row here.

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I don't want to get rid of it.

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I want to get rid of this 2 right here.

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I want to turn it into a 0.

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Let's replace this row with this row minus

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2 times that row.

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What I'm going to do is, this row minus 2

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times the first row.

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I'm going to replace this row with that.

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2 minus 2 times 1 is 0.

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That was the whole point.

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4 minus 2 times 2 is 0.

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0 minus 2 times 1 is minus 2.

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6 minus 2 times 1 is 6 minus 2, which is 4.

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4 minus 2 times 7, is 4 minus 14, which is minus 10.

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Now what can I do next.

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You can kind of see that this row, well talk more about what

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this row means.

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When all of a sudden it's all been zeroed out, there's

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nothing here.

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If I had non-zero term here, then I'd want to zero this guy

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out, although it's already zeroed out.

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I'm just going to move over to this row.

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The first thing I want to do is I want to make this leading

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coefficient here a 1.

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What I want to do is, I'm going to multiply this entire

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row by minus 1.

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If I multiply this entire row times minus 1.

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I don't even have to rewrite the matrix.

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This becomes plus 1, minus 2, plus 5.

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I think you can accept that.

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Now what can we do?

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Well, let's turn this right here into a 0.

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Let me rewrite my augmented matrix in the new form that I

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have. I'm going to keep the middle row the same this time.

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My middle row is 0, 0, 1, minus 2, and then it's

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augmented, and I get a 5 there.

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What I want to do is I want to eliminate this minus 2 here.

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Why don't I add this row to 2 times that row.

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Then I would have minus 2, plus 2, and that'll work out.

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What do I get.

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Well, these are just leading 0's.

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Then I have minus 2, plus 2 times 1.

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That's just 0.

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4 plus 2 times minus 2, that is minus 4.

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That's 4 plus minus 4, that's 0 as well.

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Then you have minus 10 plus 2 times 5.

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Well, that's just minus 10 plus 10, which is 0.

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That one just got zeroed out.

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Normally, when I just did regular elimination, I was

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happy just having the situation where I had these

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leading 1's.

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Everything below it were 0's.

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I wasn't too concerned about what was above our 1's.

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What I want to do is, I want to make

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those into a 0 as well.

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I want to make this guy a 0 as well.

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What I can do is, I can replace this first row with

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that first row minus this second row.

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What is 1 minus 0?

166
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That's just 1.

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2 minus 0 is 2.

168
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1 minus 1 is 0.

169
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1 minus minus 2 is 3.

170
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7 minus 5 is 2.

171
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There you have it.

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00:08:23,810 --> 00:08:27,970
We have our matrix in reduced row echelon form.

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This is the reduced row echelon form of our matrix,

174
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I'll write it in bold, of our matrix A right there.

175
00:08:35,950 --> 00:08:39,620
You know it's in reduced row echelon form because all of

176
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your leading 1's in each row-- so what are my

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leading 1's in each row?

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I have this 1 and I have that 1.

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They're the only non-zero entry in their columns.

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These are called the pivot entries.

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Let me label that for you.

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That's called a pivot entry.

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Pivot entry.

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They're the only non-zero entry in

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their respective columns.

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If I have any zeroed out rows, and I do have a zeroed out

187
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row, it's right there.

188
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This is zeroed out row.

189
00:09:12,659 --> 00:09:15,329
Just the style, or just the convention, is that for

190
00:09:15,330 --> 00:09:20,200
reduced row echelon form, that has to be your last row.

191
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We have the leading entries are the only -- they're all 1.

192
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That's one case.

193
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You can't have this a 5.

194
00:09:25,980 --> 00:09:29,409
You'd want to divide that equation by 5 if this was a 5.

195
00:09:29,409 --> 00:09:31,730
So your leading entries in each row are a 1.

196
00:09:31,730 --> 00:09:35,759
That the leading entry in each successive row is to the right

197
00:09:35,759 --> 00:09:38,120
of the leading entry of the row before it.

198
00:09:38,120 --> 00:09:40,289
This guy right here is to the right of that guy.

199
00:09:40,289 --> 00:09:43,399
This is just the style, the convention, of reduced row

200
00:09:43,399 --> 00:09:44,789
echelon form.

201
00:09:44,789 --> 00:09:47,629
If you have any zeroed out rows, it's in the last row.

202
00:09:47,629 --> 00:09:49,929
And finally, of course, and I think I've said this multiple

203
00:09:49,929 --> 00:09:52,859
times, this is the only non-zero entry in the row.

204
00:09:52,860 --> 00:09:54,430
What does this do for me?

205
00:09:54,429 --> 00:09:56,459
Now I can go back from this world, back

206
00:09:56,460 --> 00:09:57,850
to my linear equations.

207
00:09:57,850 --> 00:10:00,860
We remember that these were the coefficients on x1, these

208
00:10:00,860 --> 00:10:02,259
were the coefficients on x2.

209
00:10:02,259 --> 00:10:05,350
These were the coefficients on x3, on x4, and then these were

210
00:10:05,350 --> 00:10:07,000
my constants out here.

211
00:10:07,000 --> 00:10:10,730
I can rewrite this system of equations using my reduced row

212
00:10:10,730 --> 00:10:18,899
echelon form as x1, x1 plus 2x2.

213
00:10:18,899 --> 00:10:20,500
There's no x3 there.

214
00:10:20,500 --> 00:10:26,000
So plus 3x4 is equal to 2.

215
00:10:26,000 --> 00:10:29,190
This equation, no x1, no x2, I have an x3.

216
00:10:29,190 --> 00:10:38,310
I have x3 minus 2x4 is equal to 5.

217
00:10:38,309 --> 00:10:39,489
I have no other equation here.

218
00:10:39,490 --> 00:10:41,440
This one got completely zeroed out.

219
00:10:41,440 --> 00:10:45,490
I was able to reduce this system of equations to this

220
00:10:45,490 --> 00:10:47,279
system of equations.

221
00:10:47,279 --> 00:10:50,429
The variables that you associate with your pivot

222
00:10:50,429 --> 00:10:54,159
entries, we call these pivot variables.

223
00:10:54,159 --> 00:10:56,225
x1 and x3 are pivot variables.

224
00:10:56,225 --> 00:11:00,700


225
00:11:00,700 --> 00:11:02,670
The variables that aren't associated with the pivot

226
00:11:02,669 --> 00:11:05,809
entry, we call them free variables.

227
00:11:05,809 --> 00:11:12,139
x2 and x4 are free variables.

228
00:11:12,139 --> 00:11:16,539
Now let's solve for, essentially you can only solve

229
00:11:16,539 --> 00:11:17,620
for your pivot variables.

230
00:11:17,620 --> 00:11:20,370
The free variables we can set to any variable.

231
00:11:20,370 --> 00:11:21,960
I said that in the beginning of this equation.

232
00:11:21,960 --> 00:11:24,370
We have fewer equations than unknowns.

233
00:11:24,370 --> 00:11:27,940
This is going to be a not well constrained solution.

234
00:11:27,940 --> 00:11:31,170
You're not going to have just one point in R4 that solves

235
00:11:31,169 --> 00:11:31,549
this equation.

236
00:11:31,549 --> 00:11:35,019
You're going to have multiple points.

237
00:11:35,019 --> 00:11:37,860
Let's solve for our pivot variables, because that's all

238
00:11:37,860 --> 00:11:39,360
we can solve for.

239
00:11:39,360 --> 00:11:45,169
This equation tells us, right here, it tells us x3, let me

240
00:11:45,169 --> 00:11:59,959
do it in a good color, x3 is equal to 5 plus 2x4.

241
00:11:59,960 --> 00:12:12,050
Then we get x1 is equal to 2 minus x2, 2 minus 2x2.

242
00:12:12,049 --> 00:12:20,559
2 minus 2x2 plus, sorry, minus 3x4.

243
00:12:20,559 --> 00:12:24,929
I just subtracted these from both sides of the equation.

244
00:12:24,929 --> 00:12:27,629
This right here is essentially as far as we can go to the

245
00:12:27,629 --> 00:12:30,179
solution of this system of equations.

246
00:12:30,179 --> 00:12:33,729
I can pick, really, any values for my free variables.

247
00:12:33,730 --> 00:12:37,154
I can pick any values for my x2's and my x4's and I can

248
00:12:37,154 --> 00:12:38,539
solve for x3.

249
00:12:38,539 --> 00:12:40,559
What I want to do right now is write this in a slightly

250
00:12:40,559 --> 00:12:43,509
different form so we can visualize a little bit better.

251
00:12:43,509 --> 00:12:45,679
Of course, it's always hard to visualize things in four

252
00:12:45,679 --> 00:12:46,529
dimensions.

253
00:12:46,529 --> 00:12:49,839
So we can visualize things a little bit better, as to the

254
00:12:49,840 --> 00:12:51,810
set of this solution.

255
00:12:51,809 --> 00:12:53,679
Let's write it this way.

256
00:12:53,679 --> 00:12:57,259
If I were to write it in vector form, our solution is

257
00:12:57,259 --> 00:13:03,069
the vector x1, x3, x3, x4.

258
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What is it equal to?

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Well it's equal to-- let me write it like this.

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It's equal to-- I'm just rewriting, I'm just

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essentially rewriting this solution set in vector form.

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So x1 is equal to 2-- let me write a little column

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there-- plus x2.

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Let me write it this way.

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Plus x2 times something plus x4 times something.

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x1 is equal to 2 minus 2 times x2, or plus x2 minus 2.

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I put a minus 2 there.

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I can say plus x4 times minus 3.

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I can put a minus 3 there.

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This right here, the first entries of these vectors

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literally represent that equation right there. x1 is

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equal to 2 plus x2 times minus 2 plus x4 times minus 3.

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What does x3 equal?

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x3 is equal to 5.

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Put that 5 right there.

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Plus x4 times 2.

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x2 doesn't apply to it.

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We can just put a 0.

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0 times x2 plus 2 times x4.

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Now what does x2 equal?

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You could say, x2 is equal to 0 plus 1 times x2

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plus 0 times x4.

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x2 is just equal to x2.

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It's a free variable.

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Similarly, what does x4 equal to?

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x4 is equal to 0 plus 0 times x2 plus 1 times x4.

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What does this do for us?

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Well, all of a sudden here, we've expressed our solution

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set as essentially the linear combination of the linear

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combination of three vectors.

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This is a vector.

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00:15:02,039 --> 00:15:05,289
You can view it as a coordinate.

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Either a position vector.

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It is a vector in R4.

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You can view it as a position vector or a coordinate in R4.

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You could say, look, our solution set is essentially--

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this is in R4.

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Each of these have four components, but you can

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imagine it in r3.

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That my solution set is equal to some

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00:15:24,129 --> 00:15:27,179
vector, some vector there.

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That's the vector.

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Think of it is as a position vector.

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It would be the coordinate 2, 0, 5, 0.

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Which obviously, this is four dimensions right there.

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It's equal to multiples of these two vectors.

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00:15:42,159 --> 00:15:44,379
Let's call this vector, right here, let's

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call this vector a.

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00:15:46,649 --> 00:15:51,389
Let's call this vector, right here, vector b.

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00:15:51,389 --> 00:15:55,649
Our solution set is all of this point, which is right

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00:15:55,649 --> 00:15:59,439
there, or I guess we could call it that position vector.

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00:15:59,440 --> 00:16:01,590
That position vector will look like that.

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Where you're starting at the origin right there, plus

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00:16:04,559 --> 00:16:06,744
multiples of these two guys.

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00:16:06,745 --> 00:16:11,389
If this is vector a, let's do vector a in a different color.

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00:16:11,389 --> 00:16:13,330
Vector a looks like that.

321
00:16:13,330 --> 00:16:17,259
Let's say vector a looks like that, and then vector

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00:16:17,259 --> 00:16:19,620
b looks like that.

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00:16:19,620 --> 00:16:22,659
This is vector b, and this is vector a.

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00:16:22,659 --> 00:16:24,689
I don't know if this is going to be easier or harder for you

325
00:16:24,690 --> 00:16:26,720
to visualize, because obviously we are dealing in

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00:16:26,720 --> 00:16:29,340
four dimensions right here, and I'm just drawing on a two

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00:16:29,340 --> 00:16:30,910
dimensional surface.

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What you can imagine is, is that the solution set is equal

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00:16:34,259 --> 00:16:38,039
to this fixed point, this position vector, plus linear

330
00:16:38,039 --> 00:16:40,490
combinations of a and b.

331
00:16:40,490 --> 00:16:42,529
We're dealing, of course, in R4.

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Let me write that down.

333
00:16:43,309 --> 00:16:45,029
We're dealing in R4.

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00:16:45,029 --> 00:16:47,769
But linear combinations of a and b are

335
00:16:47,769 --> 00:16:50,929
going to create a plane.

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00:16:50,929 --> 00:16:54,849
You can multiply a times 2, and b times 3, or a times

337
00:16:54,850 --> 00:16:57,120
minus 1, and b times minus 100.

338
00:16:57,120 --> 00:16:59,450
You can keep adding and subtracting these linear

339
00:16:59,450 --> 00:17:01,230
combinations of a and b.

340
00:17:01,230 --> 00:17:05,960
They're going to construct a plane that contains the

341
00:17:05,960 --> 00:17:10,710
position vector, or contains the point 2, 0, 5, 0.

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00:17:10,710 --> 00:17:16,759
The solution for these three equations with four unknowns,

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is a plane in R4.

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00:17:18,660 --> 00:17:23,990


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00:17:23,990 --> 00:17:26,140
I know that's really hard to visualize, and maybe I'll do

346
00:17:26,140 --> 00:17:28,009
another one in three dimensions.

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00:17:28,009 --> 00:17:31,890
Hopefully this at least gives you a decent understanding of

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00:17:31,890 --> 00:17:35,570
what an augmented matrix is, what reduced row echelon form

349
00:17:35,569 --> 00:17:38,470
is, and what are the valid operations I can perform on a

350
00:17:38,470 --> 00:17:42,319
matrix without messing up the system.