Determine if a vector is orthogonal to the column space of a matrix transpose

Question

I am very new to linear algebra and am doing my best to catch up. I've come across the following question:

For the matrix A$\begin{bmatrix}1 & 2 & 1\\0 & 1 & 0\end{bmatrix}$ is the vector $\begin{bmatrix}1 & 0 & -1\end{bmatrix}$ orthogonal to the column space $A^T$.

In attempting this question, my thinking is that the column space of $A^T$ is the orthogonal compliment to the nullspace of A.

So, if the vector $\begin{bmatrix}1 & 0 & -1\end{bmatrix}$ is in the nullspace of A then the answer is TRUE.

On my reasoning, the nullspace of A is $\begin{bmatrix}-1 & 0 & 1\end{bmatrix}$ and so I think the answer is FALSE.

Am I thinking this through correctly? If not, any assistance would be kindly appreciated!

Answer 1

The answer is yes.

Column space of $A^{T}$ is the row space of $A$ and the given vector $$\begin{bmatrix}1 & 0 & -1\end{bmatrix}$$ is orthogonal to the rows of your matrix $A$

Answer 2

Note that

$$\operatorname{col}(A^T)=\operatorname{span}\{(1,2,1),(0,1,0\}$$

and

• $(1,0,-1)\cdot (1,2,1)=0$
• $(1,0,-1)\cdot (0,1,0)=0$

therefore the column space of $A^T$ is orthogonal to the given vector.