How do i find a basis for vector: \begin{matrix} 5 & 3 & 4 \\ 2 & 2 & -4 \\ 3 & 2 & 1 \\ -1 & 2 & 1\\ \end{matrix}
I know a basis of a vector A is a set of vectors which are linearly independent and span A. And a span of A is a set of vectors which can be linearly combined to create A
Do i just look randomly for vectors that span and are also linearly independent or is there a procedure to obtain them?
First note that your vectors are of dimension 3 so your basis set will have at most three vectors. (maybe fewer). I am assuming that your vectors are the rows.
If your vectors are the columns they are four dimensional but since you start with three your basis will have no more than three and will not be a basis for all of $R^4$.
Yes there is a method to find a linearly independent subset of vectors and to simplify them as far as possible. It is called Gauss-Jordan elimination and you can find about it in any good linear algebra text or on a resource such as Wikipedia or Wolfram.