Dirac Notation Confusion

Question

I am trying to express the vectors $0\choose 1$ and $1\choose 0$ in dirac notation wrt the basis {|$0\rangle,$|$1\rangle$}

How do I distinguish between the above two vectors given that all vectors take the form

$\lambda_{1}$|$0\rangle$ $+$ $\lambda_{2}$|$1\rangle$?

Answer 1

Hint.

You have the answer in the linear combination that you have write , interpreting $\lambda _1$ as the first component and $\lambda_2$ as the second component. $$ \lambda_1|0\rangle + \lambda_2|1\rangle=(\lambda_1,\lambda_2)^T $$

Answer 2

This is more like a comment but it's too long so I'll put this as an answer instead.

Well, it depends. There's no rule on how you put things inside the $|\dots \rangle$, sometimes it's used to denote the eigenvalue of the vector, e.g. $$ A|n\rangle=\lambda_n|n\rangle $$ .In this case $|n\rangle$ is used to denote the $n^{th}$ eigenvector of the operator $A$. Sometimes they use it to denote the spin or other properties of the quantum state like $|s, m_s\rangle$ or $|n,l\rangle$, etc.

In your case, you'll need the explicit description of $|0\rangle,|1\rangle$ to proceed. Maybe it's defined $$ {0\choose 1}=|0\rangle\ \ , {1\choose 0}=|1\rangle $$ or vise versa.