Mike's lecture on vector spaces and dual spaces: Difference between revisions
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Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors <math>\bratext{like this}</math> and to notate vectors <math>\kettext{like this}</math>. 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 <math>\bratext{like this}</math> and to notate vectors <math>\kettext{like this}</math>. Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So... | ||
<ul><li>Instead of writing covectors as <math>(x,y,z)^*</math>, I'll just write <math>\bra{x \s y \s z}</math> from now on.</li><li>Instead of writing vectors as <math>(a,b,c)</math>, I'll just write <math>\ket{a \s b \s c}</math> from now on.</li><li>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}.</li></ul> | <ul><li>Instead of writing covectors as <math>(x,y,z)^*</math>, I'll just write <math>\bra{x \s y \s z}</math> from now on.</li><li>Instead of writing vectors as <math>(a,b,c)</math>, I'll just write <math>\ket{a \s b \s c}</math> from now on.</li><li>When I want to denote the dot product of a covector and a vector, I'll write it as <math>\braket{x \s y \s z}{a \s b \s c}</math>.</li></ul> | ||
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 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?" | ||