How to find the dimension of $A$?

Question

If

$$A:=\left\{\ \begin {pmatrix} y_2 -y_3\\ y_2 \\ y_3 \end{pmatrix} \in \mathbb{C^3} \ \right\}\subseteq\mathbb{C^3}$$ how do I find the dimension of $A$. I know that $A$ is linearly independent and is a subspace of $\mathbb{C^3}$ but how can I use this to find the dimension of $A$?

Answer 1

You can see as the elements of $A$, if $$y=(y_2-y_3,y_2,y_3) \in A$$ then

$$y=(y_2-y_3,y_2,y_3)=y_2(1,1,0)+y_3(-1,0,1).$$

You can check that $$\{(1,1,0),(-1,0,1)\}$$ is a base for $A,$ thus $A$ is 2-dimensional.

Answer 2

Hint:

$A$ is the range of the matrix $$\begin{pmatrix}0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$