I suppose I would start by defining the matrices you gave in the correct fashion, by stating:
$$x,y,z,t= A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}$$
So
$$\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}= (AV_1,BV_2,CV_3,DV_3)$$
Would that be right? I am a bit weary.
That isn't coherent in any way, shape or form.
I'll just keep going, and maybe you could point out if I am wrong.
From here, my memory recollects that:
$$A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}= \begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$
For some reason, I recall setting the vector on the right to being zero in all column entries yes? Am I along the right track?
No.
I have explained I have never taken a class in these subjects, so to identify the ''thing'' you are wanting was down a certain path I learned to find them. If you are going to be awkward, forget it.
You claim you understand things about Hilbert spaces, inner products, quantum states etc, so this stuff you should be able to do. The ability to do these things is implicit when someone says they can do things in Hilbert spaces or quantum mechanics. If you can't even write down a coherent equation then it undermines all your claims you've got an understanding of Hilbert spaces or the Dirac equation.
Your mistakes are even deeper than that, the expressions you give are not even mathematically meaningful. A linear combination is the
sum of things, ie you should have written down $$A \begin{pmatrix}1\\0\\0\\0\end{pmatrix}+ B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}+C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}+D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}= \begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$, the expression $$A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}= \begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$ is meaningless.
Do you, or do you not recognize the mathematical path I was taking above? Does one not use the path I did to identify linearly independant vectors, which is a basis!!?? I know of the mathematical proceedure I used before, when I began to learn about them, if there is a much easier way to identify them, then, you are my guest. But don't sit there and patronize me for work I have never done.
Nice tactic, you spit out a few nonsense expressions and then say "Obviously you know what I mean", hoping Cpt or anyone else will think "I see how that could be reformulated into something meaningful, I'll assume he does too". No one thinks you do.
Take into respect I've never had mummy or daddy pay for freekin university class on these subjects, so a little consideration would be nice.
You can't simultaneously claim you've got familiarity with the things you claim and also say that! We're going on what
you have posted before on things like Hilbert spaces, electromagnetism, Dirac equation etc, which directly contradicts your comments like that when you show you don't know anything about vectors or polar coordinates etc.
You'll post about high level things but when someone probes your 'knowledge' at all you come up short.
/edit
And the slickest way to work out whether or not a set of vectors are linearly dependent or not is to form a matrix from their components and then compute its rank. This tells you the dimensionality of the space spanned by the vectors. If you want to know if a set of vectors form a basis then you construct the associated square matrix and compute its determinant. If its 0 then they are not a basis, they are linearly dependent. If it is non-zero then they are linearly independent.
