Orthogonal Complement of the Column Space

Question

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?

Answer 1

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.

Answer 2

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).