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Mike's lecture on vector spaces and dual spaces: Difference between revisions

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Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in __different__ spaces - one is a vector lying in a vector space, and the other is a covector lying in this new "dual space" you're learning about.
Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in __different__ spaces - one is a vector lying in a vector space, and the other is a covector lying in this new "dual space" you're learning about.
Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.


OK, so how do we use these things?
OK, so how do we use these things?
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==1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)==  
==1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)==  


One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is thus as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form (a, b, c). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.
One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number.
 
In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form \((a,b,c)\). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.
 
Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.


Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So...
Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So...
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* When I want to denote the dot product of a covector and a vector, I'll write it as \braket{x \s y \s z}{a \s b \s c}.
* When I want to denote the dot product of a covector and a vector, I'll write it as \braket{x \s y \s z}{a \s b \s c}.


Now then, let's say you're going to ask a harmlessly ordinary question like "does 81/80 vanish in 12-EDO?" But let's think: when you ask that question, what are you really asking? Another way to rephrase that question is to ask this: "if I go up four tempered 3/2's, then go down a
Now then, how do we use covectors? Or, rather, how do we recognize when our thought processes are using them, even if we never realized before these thought processes involved things called covectors? Well, let's say you're going to ask a harmlessly ordinary question like "does 81/80 vanish in 12-EDO?"
 
Now, when you ask that question, what are you really asking? For instance, another way to rephrase that question is to ask this: "if I go up four tempered 3/2's, then go down a
tempered 5/1, thus putting me at what's supposed to be 81/80 - how many steps in 12-EDO do I arrive at? Is it 0?"
tempered 5/1, thus putting me at what's supposed to be 81/80 - how many steps in 12-EDO do I arrive at? Is it 0?"


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&lt;br /&gt;
&lt;br /&gt;
Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in &lt;u&gt;different&lt;/u&gt; spaces - one is a vector lying in a vector space, and the other is a covector lying in this new &amp;quot;dual space&amp;quot; you're learning about.&lt;br /&gt;
Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in &lt;u&gt;different&lt;/u&gt; spaces - one is a vector lying in a vector space, and the other is a covector lying in this new &amp;quot;dual space&amp;quot; you're learning about.&lt;br /&gt;
&lt;br /&gt;
Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
OK, so how do we use these things?&lt;br /&gt;
OK, so how do we use these things?&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)&lt;/h2&gt;
  &lt;br /&gt;
  &lt;br /&gt;
One interesting way to think of covectors, since they're these dual vectors that &amp;quot;act on&amp;quot; normal vectors, is thus as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form (a, b, c). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.&lt;br /&gt;
One interesting way to think of covectors, since they're these dual vectors that &amp;quot;act on&amp;quot; normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number.&lt;br /&gt;
&lt;br /&gt;
In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form \((a,b,c)\). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.&lt;br /&gt;
&lt;br /&gt;
Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this &amp;quot;bra-ket&amp;quot; notation, or sometimes &amp;quot;Dirac&amp;quot; notation. So...&lt;br /&gt;
Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this &amp;quot;bra-ket&amp;quot; notation, or sometimes &amp;quot;Dirac&amp;quot; notation. So...&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on.&lt;/li&gt;&lt;li&gt;Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on.&lt;/li&gt;&lt;li&gt;When I want to denote the dot product of a covector and a vector, I'll write it as \braket{x \s y \s z}{a \s b \s c}.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on.&lt;/li&gt;&lt;li&gt;Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on.&lt;/li&gt;&lt;li&gt;When I want to denote the dot product of a covector and a vector, I'll write it as \braket{x \s y \s z}{a \s b \s c}.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
Now then, let's say you're going to ask a harmlessly ordinary question like &amp;quot;does 81/80 vanish in 12-EDO?&amp;quot; But let's think: when you ask that question, what are you really asking? Another way to rephrase that question is to ask this: &amp;quot;if I go up four tempered 3/2's, then go down a&lt;br /&gt;
Now then, how do we use covectors? Or, rather, how do we recognize when our thought processes are using them, even if we never realized before these thought processes involved things called covectors? Well, let's say you're going to ask a harmlessly ordinary question like &amp;quot;does 81/80 vanish in 12-EDO?&amp;quot;&lt;br /&gt;
&lt;br /&gt;
Now, when you ask that question, what are you really asking? For instance, another way to rephrase that question is to ask this: &amp;quot;if I go up four tempered 3/2's, then go down a&lt;br /&gt;
tempered 5/1, thus putting me at what's supposed to be 81/80 - how many steps in 12-EDO do I arrive at? Is it 0?&amp;quot;&lt;br /&gt;
tempered 5/1, thus putting me at what's supposed to be 81/80 - how many steps in 12-EDO do I arrive at? Is it 0?&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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