CHAPTER 1.2

CHAPTER 1.2: Beware of False Friends in the Matrix?

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Before delving into the intricacies of matrix multiplication, let’s look at another example to gain even better intuition about the difference between the real world and our coordinate space model. Instead of cat and rat data samples, we’ll look at the two data samples , a dangerous substance, and , which is charitably given to someone. And instead of using numbers, let’s use letters to label our entities. This means our models will resemble languages, some of which also use letters to label entities. So our first model, or language, is labeled as follows:

To an English speaker, this may look wrong, because should be called something like ‘poison’, not ‘gift’. But in German, is in fact called ‘gift’. If a German speaker tells the English speaker that they’re giving the English speaker a gift, the English speaker may be delighted because they think they’re getting . But they shouldn’t be, because what they’re ACTUALLY receiving is , which would kill them.¹

Since there is a misunderstanding, the English speaker needs to know what ‘gift’ is actually referring to; or in other words, to know the right English word to use for . So they need to translate from the language above, which resembles German, to English as follows:

gift_cob

In summary:

‘gift’ in German != ‘gift’ in English

‘gift’ in German ~ ‘poison’ in English

Relating this back to using numbers as labels:

\(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) in German != \(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) in English

\(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) in German ~ \(\begin{bmatrix} -4 \\ 1 \end{bmatrix}\) in English

Now if the English speaker tells the German speaker that they’re giving them a ‘gift’, the German speaker must translate this to a German word or expression that makes them understand that it’s . The German word for is ‘geschenk’. Going the other way around, the English word for is ‘poison’.

all_words_cob

\(\begin{bmatrix} 1/3 \\ 5/3 \end{bmatrix}\) ‘geschenk’ in German ~ \(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) ‘gift’ in English

\(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) ‘gift’ in German ~ \(\begin{bmatrix} -4 \\ 1 \end{bmatrix}\) ‘poison’ in English

But what does the label ‘poison’ mean in German? As we see in the German coordinate space, it does not point to any data sample. In fact, the label ‘poison’ does not mean anything in German. The same goes for the label ‘gescheck’ in English. Not all labels have to point to an data sample; so in some coordinate spaces, they just mean nonsense.

NEXT: CHAPTER 1.3

This is an example of a False Friend, “which is a pair of words in two different languages that look similar, but have different meanings.” Source: https://en.wikipedia.org/wiki/False_friend ↩