Any vector in the Null space of a matrix A is orthogonal to any vector in the Row space of A

Question

Let $A$ be a $m \times n$ matrix thus $A$ can be represented by the linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$

Let $v \in R(A) \implies v \in \mathbb{R}^n$

If $w \in N(A) \implies Tw=0$ I want to prove that the inner product $\langle v,w\rangle=0$ but I don't know how to proceed

Answer 1

As you know when we multiply a matrix by a vector we find the dot product of each row of the matrix and the given vector.

Since $T(w)= Aw$ and $w$ is in the null space of $T$, we have $Aw=0.$

Now in order to find $Aw$ you need to find the dot product of each row of $A$ and $w$.

Since $Aw=0$, each element $Aw$ is zero, so the dot product of each row of $A$ and $w$ must be zero.

That implies each vector in the row space is orthogonal to $w$ as well,because vectors in row space are simply linear combinations of row vectors of $A$