In one of the problem the statement is as follows:
Without computing $\mathbb{A}$, find the basis for its four fundamental subspaces: $$ \mathbb{A}= \begin{bmatrix} 1 & 0 & 0\\ 6 & 1 & 0\\ 9 & 8 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 & 4\\ 0 & 1 & 2 & 3\\ 0 & 0 & 1 & 2\end{bmatrix} $$ I understand the other parts of the solution but I do not know how the column space basis are $(1,0,0), (0,1,0),(0,0,1)$? Should they not be $(1,6,9),(0,1,8),(0,0,1)$ because the pivot columns of second matrix are the first three columns. Any help in this regard will be much appreciated. Thanks in advance.
Note that
$$\operatorname{Span}\{(1,0,0)^T, (0,1,0)^T,(0,0,1)^T\}=\mathbb{R}^3=\operatorname{Span}\{(1,6,9)^T,(0,1,8)^T,(0,0,1)^T\}$$
Hence, your answer is correct as well.