Consider the complex vector space $\mathbb{C^3}$ and the subset $$S = {(1,0,i),\ (i, 2, −3),\ \ (2 − i, 1 + i, i)}$$ Let V denote the span of S.
(a)Find a subset of S that forms a basis for V.
(b) What is the dimension of V ?
I am trying to solve this but I am not sure how to do it fully.
This is what I have thus far:
Reducing V = $\begin{bmatrix}1&i&2-i \\ 0&2&1+i \\ i&-3&i \end{bmatrix}$ to R.E.F:
$$ \begin{bmatrix}1&i&2-i \\ 0&2&1+i \\ i&-3&i \end{bmatrix} \sim \begin{bmatrix} i&-3&i \\ 0&-2i&1-i \\ 0&0&0 \end{bmatrix} $$
I know I have to somehow get the basis from this row echelon matrix but I am not sure how to proceed, please help me.
The reduced echelon form indicates that the set of vectors in $S$ are linearly dependent. Another observation is that there is no pair of vectors in $S$ such that one of them is a complex scalar multiple of another. This means any pair of vectors in $S$ is a linearly independent set and thus, forms a basis for span(S).
dimension of V is simply the number of vectors in the basis.