Dirac notation confusion I’m currently learning about quantum mechanics and vector spaces, which includes the Dirac notation. However I’m a bit confused about the following. Maybe I just don’t understand something very fundamental but any help will be appreciated.
When first introduced to the Dirac notation, I learned that a ket-vector can be represented by a column matrix of N dimensions. But after doing some more learning I noticed that people were suddenly representing these ket-vectors as the sum of their components (or the sum of these components multiplied by the orthonormal basis vectors). I am really confused by this because one of these methods gives us a matrix while the other gives us a single number.
For example, my textbook says that
$$|A\rangle = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \sum_i a_i |i\rangle$$
which seems to switch between the column matrix representation and the sum of components representation.
 A: They're the same thing. $\left( \matrix {a_1\\a_2\\a_3}\right)$ is the same as $a_1 \left( \matrix {1\\0\\0}\right)+a_2 \left( \matrix{0\\1\\0}\right)+a_3 \left( \matrix{0\\0\\1}\right)$
A: The answer above is perfectly good. I'll just add that sometimes, people treat the two equations you wrote slightly differently.
Suppose you're navigating with somebody and you need to talk about directions. You could say, "hey, remember that $(0, 1)$ is North and $(1, 0)$ is East" and then just talk in terms of numbers, like "head towards $(4, 6)$". But from a certain standpoint, this isn't a good idea because a compass direction is just not the same thing as a list of two numbers.
To turn a direction into a list of two numbers, you need to make an arbitrary choice, i.e. you could have associated $(1, 0)$ with North instead. And nobody will know what you're talking about unless they know what arbitrary choice you made. That is, the equation
$$\text{northeast} = \text{north} + \text{east}$$
is perfectly unambiguous, but
$$\text{northeast} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$
depends on your particular conventions.
So after a bit of experience with Dirac notation, we prefer to stop writing expressions like the second equation, because of these problems. Instead, we keep everything in terms of the abstract vectors themselves. The analogue to writing "$\text{northeast} = \text{north} + \text{east}$" is
$$|A \rangle = \sum_i a_i |i \rangle$$
and this is perfectly unambiguous. The only time we typically go back to column notation is when it's obvious what the coordinate system we're using is. (In this case, it is the coordinate system where the basis vectors are the $|i \rangle$ vectors themselves.)
