쉐도잉 연습: Independence, Basis, and Dimension - YouTube로 영어 말하기 배우기

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So as long as I'm introducing the idea of a vector space,
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So as long as I'm introducing the idea of a vector space,
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I better introduce the things that go with it,
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the idea of its dimension,
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and all important, the idea of a basis for that space.
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That space could be all of three-dimensional space,
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the space we live in.
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In that case, the dimension is three.
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But what's the meaning of a basis,
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A basis for three-dimensional space.
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Or a basis for other spaces.
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OK, so I have to explain independence, basis, and dimension.
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Dimension's easy if you get the first two.
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OK, independence.
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Are those vectors independent?
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Well, if I draw them in three-dimensional space,
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I can imagine 215 going in some direction.
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Let me draw it.
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How's that?
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2, 1, 5, whatever.
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Grows there.
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That's A1.
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OK.
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Now, is A2 on the same line?
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If A2 is on the same line,
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then it would be dependent.
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The two vectors would be dependent if they're on the same line.
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But this one is not on that line.
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It's 4, 2, 0.
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So it doesn't go up at all.
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It's somewhere in this plane, 4, 2, 0.
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I'll say there.
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Whatever.
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A2.
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So those are independent.
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Now, where does, so their combinations give me a space.
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The combinations of A1 and A2 give me a plane,
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a flat plane, in three-dimensional space.
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That plane is, I would say, they span the plane.
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A1 and A2 span a plane.
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And here's the key word, span.
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So there are two vectors.
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They're in three-dimensional space.
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And the plane they span is all their combinations.
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That's what we're always doing,
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taking all the combinations of these vectors.
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OK.
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So actually, A1 and A2 are a basis for that plane.
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A1 and A2 are a basis for that plane because their combinations fill the plane.
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And also, they're independent.
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I need them both.
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If I throw away one,
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I would only have one vector left,
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and it would only span a line.
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OK, now let me bring in a third vector in three dimensions.
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Well, what shall I take for that third vector?
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Suppose I take a1 plus a2 as my third vector.
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So 6, 3, 5.
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What about the vector 6, 3, 5?
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Well, what do I know?
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It's obviously special.
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It's a1 plus a2.
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It's in the same plane.
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So if I took a3 equal 6,
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3, 5, that would be dependent.
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The three vectors would be dependent with that a3,
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they would span the plane still.
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Their combinations would still give the plane.
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But they wouldn't be a basis for the plane a1 and a2 and a3 together,
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that's too much.
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Too many vectors for a single plane.
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The vectors are dependent.
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And we don't, a basis has to be independent vectors.
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You have to need them all.
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We don't need all three here.
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So that's a dependent one.
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It can't go into a basis with a1 and a2,
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because the three vectors are dependent.
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Now let me make a different choice.
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So that was dead.
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That did not do it.
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All right, let me take a3 equal some other,
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not a combination of these,
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but headed off in some new direction.
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Well, I don't know what that new direction is.
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Maybe 1, 0, 0.
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What the heck?
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I believe, I hope I'm right,
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that 1, 0, 0 is not a combination here.
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I think 1, 0, 0 goes off.
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It's pretty short.
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There's a3.
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Better a3 than that loser 6, 3, 5.
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1, 0, 0 is a winner.
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These three vectors.
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So now a1, a2, and let me add in a3.
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all three of them span a.
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What do they span?
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What are all the combinations of a1, a2, a3?
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It's three-dimensional.
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It's the whole three-dimensional space.
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They span all of 3D, the whole three-dimensional space.
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They're a basis for the whole three-dimensional space.
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They're independent.
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So let me, do you see that picture before I move it?
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A1, A2, A3 are independent.
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None of them is a combination of the others.
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They fill a three-dimensional space.
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They're a basis for that three-dimensional space.
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And that space is the whole,
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in this example, is the whole of R3.
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So let me just write down on the next backward what I mean.
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Independent.
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Independent.
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So independent columns of a matrix,
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independent columns of a matrix A,
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means the only solution to A v equals 0 is v equals 0.
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So if I have independent columns,
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then I haven't got any null space.
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If I have independent columns,
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then the null space of the matrix is just the 0 vector,
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is the 0 vector.
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So let me write down that example again.
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A was the matrix 215, 420, 100.
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I believe that matrix has independent columns.
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So its column space is the full three-dimensional space.
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Its null space only contains,
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let me put it, make that clear that that's a vector.
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And now I'm ready to explain,
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write down the idea of a basis.
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So what is a basis for the space?
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A basis for a space, a subspace.
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Independent vectors, that's the key.
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independent vectors that span the space,
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the subspace, whatever it is.
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By the way, if the column space is all of three-dimensional space,
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as it is here, That's a subspace too.
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It's the whole space, but the whole space counts as a subspace of itself.
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And the 0 vector alone counts as the smallest possible.
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So if we're in three dimensions,
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the idea of subspaces has we have just the 0 vector, just one point.
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That's the smallest.
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We have the whole three-dimensional space.
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That's the biggest.
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And then we have all the lines through 0.
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Those are on the small side.
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We have all the planes through 0.
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Those are a bit bigger.
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So we have, and those dimensions are 0, 1, 2, 3.
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The possible dimensions is told to us by how many basis vectors we need.
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So let me look at that and then come to dimension.
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OK.
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So independent means that no other combination of the vectors,
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no combination of these vectors gives the 0 vector except to take 0 of that,
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0 of that, and 0 of that.
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Basis, so those are a basis for the column space because they're independent and their combinations give the whole column space.
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OK.
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And now I wanted to say something about dimensions.
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OK.
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Dimension.
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It's a number.
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It's the number of basis vectors.
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for the subspace.
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Oh, but you might say that the subspace has other bases,
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not just the one you happen to think of first.
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And I agree.
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Many different bases.
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All I need for this example,
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all I need to get a basis for,
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in this case, for three-dimensional space is I need three independent vectors, any three.
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But the point is, the point about dimension is that I need exactly three.
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I can never get two vectors that span all of R3.
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And I can never get four vectors that are independent in R3.
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If I have fewer than the dimension number, I don't have enough.
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They don't span.
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If I have too many than the dimension, they're dependent.
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They won't be independent.
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They can't be a basis.
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Every basis has the same number,
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and that number is the dimension of the subspace.
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All right.
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Let's just take an example, just with a picture.
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I'll stay in three-dimensional space,
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but my subspace will just be a plane.
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So here I'm in three-dimensional space.
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Good.
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Now I have my subspace is a plane.
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So it goes through the origin.
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But it's only a plane.
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So I'm expecting that I could take a vector in the plane.
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And I could take another vector in the plane.
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And they could be independent.
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They are.
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They're different directions.
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But I couldn't find a third independent vector in a plane.
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Every basis for the plane.
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So here, every basis for this plane
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contains two vectors, always two.
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And that number two is the dimension of a plane.
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Well, I'm just saying the plane there is two dimensional.
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It's not the same as R2.
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It's not the same.
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That plane is a plane in R3.
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It's not ordinary two-dimensional space.
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But its dimension is two,
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because it takes any vector.
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And if I didn't like the looks of this one,
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well, that's no problem.
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Let me go that way.
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That's just as good.
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Those two vectors are independent.
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They span the plane.
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They're a basis for the plane.
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The plane is two-dimensional.
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That's the set of key ideas.
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Independent, span, basis.
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Basis is fundamental.
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Basis is a bunch of vectors.
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And dimension is how many vectors.
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OK, those are key ideas in linear algebra.
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And you'll see them come into the big picture of linear algebra.
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Thank you.
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Thank you.

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