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Title: Outer Product
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Series: Advent of Mathematical Symbols
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YouTube-Title: Outer Product
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Bright video: https://youtu.be/1CCW_VJM7Ao
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Dark video: https://youtu.be/AdD2ChAu5Vo
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: aoms18_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,629 –> 00:00:01,969 Hello and welcome.
2 00:00:02,169 –> 00:00:06,300 The mathematical symbol of today is given by the outer product for vectors.
3 00:00:06,371 –> 00:00:09,226 Which is given by this product sign here.
4 00:00:09,943 –> 00:00:14,059 So you see it’s a common multiplication sign with a circle around it.
5 00:00:14,800 –> 00:00:20,643 Now since this is such a nice symbol it’s also often used to denote other mathematical objects.
6 00:00:21,471 –> 00:00:25,845 However the outer product for vectors might be the simplest one of these.
7 00:00:26,543 –> 00:00:31,814 Indeed this outer product here is related to the kronecker product for matrices.
8 00:00:31,871 –> 00:00:34,125 Which is also denoted with this symbol.
9 00:00:35,143 –> 00:00:40,536 Therefore some people say the outer product is just a kronecker product, but for vectors.
10 00:00:41,229 –> 00:00:44,329 So maybe i can immediately explain it.
11 00:00:44,486 –> 00:00:47,356 We have one vector on the left hand side of the sign
12 00:00:47,357 –> 00:00:48,798 and one on the right hand side.
13 00:00:49,686 –> 00:00:51,889 Indeed the dimensions could be different.
14 00:00:52,089 –> 00:00:56,592 So we have the 2-dimensional vector here and maybe a 3-dimensional vector on the right hand side.
15 00:00:57,586 –> 00:01:01,733 In other words we have a vector v times a vector w.
16 00:01:02,814 –> 00:01:06,005 and now what should come out is a matrix.
17 00:01:06,829 –> 00:01:12,471 Indeed the height of the matrix is given by the vector v and the width is given by the vector w.
18 00:01:13,614 –> 00:01:17,206 Now the entries of the matrix are not so hard to remember,
19 00:01:17,406 –> 00:01:22,303 because in each column of the matrix we first find a copy of the vector v.
20 00:01:22,986 –> 00:01:29,197 Of course the vector w comes also in, but now we find a copy of it in each row.
21 00:01:30,314 –> 00:01:33,356 This means that you first have to transpose the vector.
22 00:01:33,557 –> 00:01:35,700 and then you can push it to the row.
23 00:01:36,643 –> 00:01:39,300 and in the same way we have it for the second row.
24 00:01:40,629 –> 00:01:47,073 Ok and now you see in each entry of the matrix we just have the product for ordinary numbers.
25 00:01:48,300 –> 00:01:53,573 In fact this is already the whole construction for the kronecker product with two vectors.
26 00:01:54,371 –> 00:01:59,771 and of course it’s not hard at all to generalize this to other dimensions than 2 and 3.
27 00:02:00,786 –> 00:02:04,471 You just have to remember how the matrix entries are formed.
28 00:02:05,386 –> 00:02:09,043 So v times w gives us always a matrix.
29 00:02:09,900 –> 00:02:15,051 and now we can just look at a general entry with indices “i” and j.
30 00:02:15,729 –> 00:02:23,139 and as you have learned before this one is simply given by v_i times w_j.
31 00:02:23,943 –> 00:02:28,428 So you see we can use this for the definition of the outer product.
32 00:02:29,171 –> 00:02:32,176 and with this i hope you have learned something today.
33 00:02:32,900 –> 00:02:34,515 Then see you next time.
34 00:02:34,715 –> 00:02:35,686 Bye!
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Quiz Content
Q1: Let $v,w \in \mathbb{R}^2$ be two vectors. What is always correct for $ v \otimes w$?
A1: $v \otimes w \in \mathbb{R}^{2 \times 2}$
A2: $v \otimes w = w \otimes v$
A3: $v \otimes w \in \mathbb{R}^{4}$
A4: $v \otimes w + w \otimes v = 0$
Q2: Let $v \in \mathbb{R}$ and $w \in \mathbb{R}^n$. What is always correct for $ v \otimes w$?
A1: $v \otimes w \in \mathbb{R}^{1 \times n}$
A2: $v \otimes w = w $
A3: $v \otimes w \in \mathbb{R}^{n}$
A4: $v \otimes w = v $
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Last update: 2024-11
