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

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 ==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 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.
 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 instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on. And instead of writing vectors as \((a,b,c)), I'll just write \(\ket{|a \s b \s c&gt;}\) from now on.
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-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 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;
 &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 instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on. And instead of writing vectors as \((a,b,c)), I'll just write \(\ket{|a \s b \s c&amp;gt;}\) from now on.&lt;br /&gt;