Pratica di Shadowing: Independence, Basis, and Dimension - Impara a parlare inglese con YouTube

C2
So as long as I'm introducing the idea of a vector space,
⏸ In Pausa
222 frasi
Se le frasi sono troppo corte o troppo lunghe, clicca su Edit per modificarle.
1
So as long as I'm introducing the idea of a vector space,
2
I better introduce the things that go with it,
3
the idea of its dimension,
4
and all important, the idea of a basis for that space.
5
That space could be all of three-dimensional space,
6
the space we live in.
7
In that case, the dimension is three.
8
But what's the meaning of a basis,
9
A basis for three-dimensional space.
10
Or a basis for other spaces.
11
OK, so I have to explain independence, basis, and dimension.
12
Dimension's easy if you get the first two.
13
OK, independence.
14
Are those vectors independent?
15
Well, if I draw them in three-dimensional space,
16
I can imagine 215 going in some direction.
17
Let me draw it.
18
How's that?
19
2, 1, 5, whatever.
20
Grows there.
21
That's A1.
22
OK.
23
Now, is A2 on the same line?
24
If A2 is on the same line,
25
then it would be dependent.
26
The two vectors would be dependent if they're on the same line.
27
But this one is not on that line.
28
It's 4, 2, 0.
29
So it doesn't go up at all.
30
It's somewhere in this plane, 4, 2, 0.
31
I'll say there.
32
Whatever.
33
A2.
34
So those are independent.
35
Now, where does, so their combinations give me a space.
36
The combinations of A1 and A2 give me a plane,
37
a flat plane, in three-dimensional space.
38
That plane is, I would say, they span the plane.
39
A1 and A2 span a plane.
40
And here's the key word, span.
41
So there are two vectors.
42
They're in three-dimensional space.
43
And the plane they span is all their combinations.
44
That's what we're always doing,
45
taking all the combinations of these vectors.
46
OK.
47
So actually, A1 and A2 are a basis for that plane.
48
A1 and A2 are a basis for that plane because their combinations fill the plane.
49
And also, they're independent.
50
I need them both.
51
If I throw away one,
52
I would only have one vector left,
53
and it would only span a line.
54
OK, now let me bring in a third vector in three dimensions.
55
Well, what shall I take for that third vector?
56
Suppose I take a1 plus a2 as my third vector.
57
So 6, 3, 5.
58
What about the vector 6, 3, 5?
59
Well, what do I know?
60
It's obviously special.
61
It's a1 plus a2.
62
It's in the same plane.
63
So if I took a3 equal 6,
64
3, 5, that would be dependent.
65
The three vectors would be dependent with that a3,
66
they would span the plane still.
67
Their combinations would still give the plane.
68
But they wouldn't be a basis for the plane a1 and a2 and a3 together,
69
that's too much.
70
Too many vectors for a single plane.
71
The vectors are dependent.
72
And we don't, a basis has to be independent vectors.
73
You have to need them all.
74
We don't need all three here.
75
So that's a dependent one.
76
It can't go into a basis with a1 and a2,
77
because the three vectors are dependent.
78
Now let me make a different choice.
79
So that was dead.
80
That did not do it.
81
All right, let me take a3 equal some other,
82
not a combination of these,
83
but headed off in some new direction.
84
Well, I don't know what that new direction is.
85
Maybe 1, 0, 0.
86
What the heck?
87
I believe, I hope I'm right,
88
that 1, 0, 0 is not a combination here.
89
I think 1, 0, 0 goes off.
90
It's pretty short.
91
There's a3.
92
Better a3 than that loser 6, 3, 5.
93
1, 0, 0 is a winner.
94
These three vectors.
95
So now a1, a2, and let me add in a3.
96
all three of them span a.
97
What do they span?
98
What are all the combinations of a1, a2, a3?
99
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.

Scarica l'app

Valutazione AI per ogni frase che pronunci

Scansiona per scaricare
Scansiona per scaricare
Download on theApp Store
Get it onGoogle Play
TRENDING

Popolari

Perché praticare il parlare con questo video?

Questo video è un'ottima risorsa per chi vuole praticare la conversazione in inglese in un contesto matematico ma accessibile. La spiegazione di concetti come "indipendenza", "base" e "dimensione" viene presentata in modo chiaro e coinvolgente, permettendo agli studenti di assimilare la terminologia tecnica e la struttura della lingua inglese. Praticare la lingua in contesti diversi, come nei video, migliora non solo la comprensione ma anche la capacità di utilizzare l'inglese in conversazioni quotidiane. Utilizzando tecniche come shadow speech, puoi ripetere immediatamente ciò che ascolti, aiutandoti a perfezionare la tua pronuncia e il tuo accento.

Grammatica & Espressioni nel Contesto

Analizziamo alcune delle strutture grammaticali chiave utilizzate nel video:

  • Domande retoriche: “Are those vectors independent?” Questa struttura invita gli ascoltatori a riflettere attivamente su ciò che viene detto, incoraggiando una partecipazione attiva.
  • Uso del futuro semplice: “I’ll say there.” Qui l’oratore utilizza il futuro semplice per fare previsioni o affermazioni, che è una pratica comune nei discorsi informali.
  • Espressioni relative all'indipendenza: “If A2 is on the same line, then it would be dependent.” Questa struttura aiuta a collegare le idee e a chiarire i significati attraverso condizioni.

Queste strutture possono essere modelli preziosi quando si impara l'inglese con YouTube, poiché forniscono un esempio pratico di come usare il linguaggio in contesti matematici e tecnici.

Trappole Comuni per la Pronuncia

Durante l'ascolto del video, fai attenzione a queste parole e accenti che possono confondere gli studenti:

  • Indipendenza: La pronuncia corretta di “independence” può risultare difficile a causa delle sillabe forti e deboli. Esercitati a pronunciarla enfatizzando le sillabe appropriate.
  • Base: In inglese, “basis” è spesso confuso con “basis” (pronunciato “beɪsɪs”), a causa della somiglianza con il termine italiano “base”. Assicurati di ripetere questi termini utilizzando shadow speech per migliorare la tua pronuncia inglese.
  • Dimensione: La parola “dimension” può offrire difficoltà, specialmente nel passaggio dei suoni e nelle intonazioni. Unita a vocaboli descrittivi come "three-dimensional," diventa essenziale perfezionare la pronuncia per comunicazioni più fluide.

Affrontando queste trappole di pronuncia, non solo migliorerai la tua capacità di comunicare ma anche la tua fiducia nel parlare inglese.

Cos'è la tecnica dello Shadowing?

Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.

☕ Offrici un caffè