column vector times row vector - Transpose

Question

is that always true for two nx1 Vectors? Where x is a column vector of (x,y) and a a column vector of (a,b).

$(x-a)^T *grad(f(a) = grad(f(a))^T*(x-a)$

Answer 1

Yes. (Assuming your $*$ means matrix multiplication.)

Take two row vectors $a,b \in \mathbb R ^ {n \times 1}$. Actually, they are row matrices. Denote the enties of these matrices as $\{a_i\}_{i=1,\dots,n},\{b_i\}_{i=1,\dots,n}$. Then we have

$a^T b = \sum_{i=1}^n a_i b_i = \sum_{i=1}^n b_i a_i = b^T a,$

where in the first and last $=$ the definition of matrix multiplication was used.

We just showed that the scalar product of two row vectors is commutative: $(a,b)=(b,a)$.