Is the statement $v^Tw =w^Tv$ true?

Question

I have two vectors $v,w \in \mathbb{R}^n$. Is the statement $v^Tw =w^Tv$ true?

In my opinion yes, because both are a single number, and this should be the same.

Let $v:= \left( \begin{matrix} x\\y\\z \end{matrix} \right)$ and $w:=\left( \begin{matrix} a\\b\\c \end{matrix} \right) $.

Then we get $$v^Tw = \left( \begin{matrix} x & y & z \end{matrix} \right) \left( \begin{matrix} a \\b \\c \end{matrix} \right) = xa+yb+zc = ax+by+cz = \left( \begin{matrix} a&b&c \end{matrix} \right) \left( \begin{matrix} x\\y\\z \end{matrix} \right)= w^Tv$$ because the multiplication is commutative.

Is this ok?

Answer 1

Yes, it is correct :). This is actually a way to define the (usual) dot product in $\mathbb{R}^n$ : $v\cdot w := v^T w = w^T v$. (so remember this identity! It comes up a lot in linear algebra ^^)

Answer 2

Yes it is ok. Your apprach is right . It is just the commutativity of $\mathbb R$.