Orthogonal Complement of the Column Space I currently have this problem with this matrix.

Of this matrix i have to calculate the Orthogonal Complement of the Column Space. But nothing is given?
How can you do this?
Thank you in advantage.
 A: The orthogonal complement of $\textrm{Col}(A)$ is $\textrm{Nul}(A^T)$. 
The orthogonal complement of $\textrm{Col}(A)$ is the set of vectors $\vec{z}$ that are orthogonal to $\textit{each}$ vector in $\textrm{Col}(A)$, i.e. each vector given by $A\vec{x}$ for any $\vec{x} \in \mathbb{R}^n$. That means for each $\vec{x}$, we have $A\vec{x} \cdot \vec{z} = 0$. Using the definition of the dot product, $\vec{u} \cdot \vec{v} = \vec{u}^T \vec{v}$, this can be written as 
$$(A\vec{x})^T \vec{z} = 0$$
Then using the fact that $(XY)^T = Y^TX^T$, we can rewrite this as 
$$ \vec{x}^TA^T \vec{z} = 0$$
Since we need this to hold for $\textit{any}$ $\vec{x}$, we need $A^T\vec{z} = 0$ meaning any $\vec{z} \in \textrm{Nul}(A^T)$ is in the orthogonal complement of $\textrm{Col}(A)$
A: You want those vectors which are orthogonal to the columns of given matrix $A$. So you want a row vector $x$ such that
$$xA=0$$
Taking transpose
$$A^Tx^T=0$$
So essentially you want to solve for the solution of the homogeneous system with the matrix $A^T$.
note: this exercise is based on the fact that the null space is orthogonal to row space (same as column space of the transposed matrix).
A: Given a matrix $A$, you are looking for the space of all vectors $v$ that are orthogonal to the span of the columns. I would try to find an expression for that, using the fact that the being orthogonal to the column space is the same as being orthogonal to all the columns.
