تدريب Shadowing: Independence, Basis, and Dimension - تعلم التحدث بالإنجليزية مع YouTube

So as long as I'm introducing the idea of a vector space,

⏸ متوقف مؤقتاً

السرعة:

222 جمل

إذا كانت الجمل قصيرة أو طويلة جدًا، انقر على Edit لتعديلها.

So as long as I'm introducing the idea of a vector space,

I better introduce the things that go with it,

the idea of its dimension,

and all important, the idea of a basis for that space.

That space could be all of three-dimensional space,

the space we live in.

In that case, the dimension is three.

But what's the meaning of a basis,

A basis for three-dimensional space.

Or a basis for other spaces.

OK, so I have to explain independence, basis, and dimension.

Dimension's easy if you get the first two.

OK, independence.

Are those vectors independent?

Well, if I draw them in three-dimensional space,

I can imagine 215 going in some direction.

Let me draw it.

How's that?

2, 1, 5, whatever.

Grows there.

That's A1.

OK.

Now, is A2 on the same line?

If A2 is on the same line,

then it would be dependent.

The two vectors would be dependent if they're on the same line.

But this one is not on that line.

It's 4, 2, 0.

So it doesn't go up at all.

It's somewhere in this plane, 4, 2, 0.

I'll say there.

Whatever.

A2.

So those are independent.

Now, where does, so their combinations give me a space.

The combinations of A1 and A2 give me a plane,

a flat plane, in three-dimensional space.

That plane is, I would say, they span the plane.

A1 and A2 span a plane.

And here's the key word, span.

So there are two vectors.

They're in three-dimensional space.

And the plane they span is all their combinations.

That's what we're always doing,

taking all the combinations of these vectors.

OK.

So actually, A1 and A2 are a basis for that plane.

A1 and A2 are a basis for that plane because their combinations fill the plane.

And also, they're independent.

I need them both.

If I throw away one,

I would only have one vector left,

and it would only span a line.

OK, now let me bring in a third vector in three dimensions.

Well, what shall I take for that third vector?

Suppose I take a1 plus a2 as my third vector.

So 6, 3, 5.

What about the vector 6, 3, 5?

Well, what do I know?

It's obviously special.

It's a1 plus a2.

It's in the same plane.

So if I took a3 equal 6,

3, 5, that would be dependent.

The three vectors would be dependent with that a3,

they would span the plane still.

Their combinations would still give the plane.

But they wouldn't be a basis for the plane a1 and a2 and a3 together,

that's too much.

Too many vectors for a single plane.

The vectors are dependent.

And we don't, a basis has to be independent vectors.

You have to need them all.

We don't need all three here.

So that's a dependent one.

It can't go into a basis with a1 and a2,

because the three vectors are dependent.

Now let me make a different choice.

So that was dead.

That did not do it.

All right, let me take a3 equal some other,

not a combination of these,

but headed off in some new direction.

Well, I don't know what that new direction is.

Maybe 1, 0, 0.

What the heck?

I believe, I hope I'm right,

that 1, 0, 0 is not a combination here.

I think 1, 0, 0 goes off.

It's pretty short.

There's a3.

Better a3 than that loser 6, 3, 5.

1, 0, 0 is a winner.

These three vectors.

So now a1, a2, and let me add in a3.

all three of them span a.

What do they span?

What are all the combinations of a1, a2, a3?

It's three-dimensional.

100

It's the whole three-dimensional space.

101

They span all of 3D, the whole three-dimensional space.

102

They're a basis for the whole three-dimensional space.

103

They're independent.

104

So let me, do you see that picture before I move it?

105

A1, A2, A3 are independent.

106

None of them is a combination of the others.

107

They fill a three-dimensional space.

108

They're a basis for that three-dimensional space.

109

And that space is the whole,

110

in this example, is the whole of R3.

111

So let me just write down on the next backward what I mean.

112

Independent.

113

Independent.

114

So independent columns of a matrix,

115

independent columns of a matrix A,

116

means the only solution to A v equals 0 is v equals 0.

117

So if I have independent columns,

118

then I haven't got any null space.

119

If I have independent columns,

120

then the null space of the matrix is just the 0 vector,

121

is the 0 vector.

122

So let me write down that example again.

123

A was the matrix 215, 420, 100.

124

I believe that matrix has independent columns.

125

So its column space is the full three-dimensional space.

126

Its null space only contains,

127

let me put it, make that clear that that's a vector.

128

And now I'm ready to explain,

129

write down the idea of a basis.

130

So what is a basis for the space?

131

A basis for a space, a subspace.

132

Independent vectors, that's the key.

133

independent vectors that span the space,

134

the subspace, whatever it is.

135

By the way, if the column space is all of three-dimensional space,

136

as it is here, That's a subspace too.

137

It's the whole space, but the whole space counts as a subspace of itself.

138

And the 0 vector alone counts as the smallest possible.

139

So if we're in three dimensions,

140

the idea of subspaces has we have just the 0 vector, just one point.

141

That's the smallest.

142

We have the whole three-dimensional space.

143

That's the biggest.

144

And then we have all the lines through 0.

145

Those are on the small side.

146

We have all the planes through 0.

147

Those are a bit bigger.

148

So we have, and those dimensions are 0, 1, 2, 3.

149

The possible dimensions is told to us by how many basis vectors we need.

150

So let me look at that and then come to dimension.

151

OK.

152

So independent means that no other combination of the vectors,

153

no combination of these vectors gives the 0 vector except to take 0 of that,

154

0 of that, and 0 of that.

155

Basis, so those are a basis for the column space because they're independent and their combinations give the whole column space.

156

OK.

157

And now I wanted to say something about dimensions.

158

OK.

159

Dimension.

160

It's a number.

161

It's the number of basis vectors.

162

for the subspace.

163

Oh, but you might say that the subspace has other bases,

164

not just the one you happen to think of first.

165

And I agree.

166

Many different bases.

167

All I need for this example,

168

all I need to get a basis for,

169

in this case, for three-dimensional space is I need three independent vectors, any three.

170

But the point is, the point about dimension is that I need exactly three.

171

I can never get two vectors that span all of R3.

172

And I can never get four vectors that are independent in R3.

173

If I have fewer than the dimension number, I don't have enough.

174

They don't span.

175

If I have too many than the dimension, they're dependent.

176

They won't be independent.

177

They can't be a basis.

178

Every basis has the same number,

179

and that number is the dimension of the subspace.

180

All right.

181

Let's just take an example, just with a picture.

182

I'll stay in three-dimensional space,

183

but my subspace will just be a plane.

184

So here I'm in three-dimensional space.

185

Good.

186

Now I have my subspace is a plane.

187

So it goes through the origin.

188

But it's only a plane.

189

So I'm expecting that I could take a vector in the plane.

190

And I could take another vector in the plane.

191

And they could be independent.

192

They are.

193

They're different directions.

194

But I couldn't find a third independent vector in a plane.

195

Every basis for the plane.

196

So here, every basis for this plane

197

contains two vectors, always two.

198

And that number two is the dimension of a plane.

199

Well, I'm just saying the plane there is two dimensional.

200

It's not the same as R2.

201

It's not the same.

202

That plane is a plane in R3.

203

It's not ordinary two-dimensional space.

204

But its dimension is two,

205

because it takes any vector.

206

And if I didn't like the looks of this one,

207

well, that's no problem.

208

Let me go that way.

209

That's just as good.

210

Those two vectors are independent.

211

They span the plane.

212

They're a basis for the plane.

213

The plane is two-dimensional.

214

That's the set of key ideas.

215

Independent, span, basis.

216

Basis is fundamental.

217

Basis is a bunch of vectors.

218

And dimension is how many vectors.

219

OK, those are key ideas in linear algebra.

220

And you'll see them come into the big picture of linear algebra.

221

Thank you.

222

Thank you.

TRENDING

الأكثر شعبية

ما هي تقنية التظليل الصوتي؟

التظليل الصوتي (Shadowing) تقنية تعلم لغة مدعومة علمياً، طُورت أصلاً لتدريب المترجمين الفوريين المحترفين. الطريقة بسيطة لكنها قوية: تستمع لصوت إنجليزي أصلي وتكرره فوراً بصوت عالٍ — كظل يتبع المتحدث بتأخير 1-2 ثانية. تُظهر الأبحاث تحسناً كبيراً في دقة النطق والتنغيم والإيقاع وربط الأصوات والاستماع والطلاقة.

☕ اشترِ لنا قهوة

يظل ShadowingEnglish مجانيًا بنسبة 100% بفضل دعمكم. تكاليف الخوادم والذكاء الاصطناعي مرتفعة — قهوتكم تبقينا مستمرين! 🙏

التبرع عبر PayPal

ShadowingEnglish.com – تمرين التظليل الصوتي للإنجليزية

تحدث الإنجليزية بطلاقة باستخدام تقنية التظليل الصوتي. استمع لفيديوهات يوتيوب الأصلية، كرر جملة بجملة، وابنِ نطقاً حقيقياً وطلاقة — يستخدمها متعلمو IELTS حول العالم.

تدريب Shadowing: Independence, Basis, and Dimension - تعلم التحدث بالإنجليزية مع YouTube

الأكثر شعبية

لماذا ممارسة المحادثة مع هذا الفيديو؟

القواعد والتعابير في السياق

فخاخ النطق الشائعة

ما هي تقنية التظليل الصوتي؟

تدريب Shadowing: Independence, Basis, and Dimension - تعلم التحدث بالإنجليزية مع YouTube

تنزيل التطبيق

لماذا ممارسة المحادثة مع هذا الفيديو؟

القواعد والتعابير في السياق

فخاخ النطق الشائعة

ما هي تقنية التظليل الصوتي؟