Question

Say v1 = [1 1 2]', v2 = [2 2 5]' and v3 = [3 3 8]'. Put them together and we have a rank 2 matrix, as there are 2 pivot columns in R. However, it seems its implied the 3 vectors are not independent as we can have a columnspace of dim 2 = r. If the 3 vectors are dependent then it seems that it should be possible to express v3 as a linear combination of v1 and v2. Is that right?

Answer 1

The matrix formed by those vectors is not rank 2, but rank 3. Is it possible that you made a mistake in typing the vectors? If it were true, your reasoning is mostly correct though.

Answer 2

The rank is 2. The reduced row echelon form of [1 2 3; 1 2 3; 2 5 8] is [1 0 -1; 0 1 2; 0 0 0]. There are 2 pivot columns and 1 free column.

Answer 3

I take that back. I must be tired. Either way you form them generates a matrix of rank 2. You are correct. -1*v1 + 2*v2 gives you v3.

Answer 4

Yes, you are right. You can get the answer by solving the equation below.
\[\left[\begin{matrix}1 & 2 \\ 1 & 2\\ 2 & 5\end{matrix}\right]\left[\begin{matrix} a \\ b\end{matrix}\right]=\left[\begin{matrix}3 \\ 3\\ 8\end{matrix}\right]\]