Are the following vectors in the range of A

Question

Let $A$ be the following matrix:

$$ \left( \begin{array}{cccccc} 1 & 2 & 1 & 3 & 2 & 1\\ 2 & 0 & 3 & 2 & 3 & 0 \\ 4 & 2 & 1 & 1 & 2 & 1 \\ 1 & 0 & -3 & -4 & -3 & 0 \end{array} \right) $$

I showed that the rank of the matrix is 3. However, I was wondering, is there an elegant way to check whether the vectors $(1,4,2)^T$,$(1,3,6,2)^T$ and $(0,0,0,1)^T$ are in the range of the matrix.

Thanks in advance.

Answer 1

Since $A$ is $4\times 6$, the domain is $\mathbb{R}^6$ and the range is a subspace of $\mathbb{R}^4$.

Your first vector listed is not in $\mathbb{R}^4$, so cannot be in the range.

For the other 2, the standard way to check would be to augment $A$ with each vector in turn, reduce, and see if the system has a solution (in this case, that would mean the bottom row of the reduced matrix is entirely zeros).

Note that to reduce the amount of effort, you only really need to check if the vectors are in the span of the 3 independent columns of $A$ (those corresponding to the pivot columns). The others are just redundant and can be thrown away.