Am I doing something wrong with matrix multiplication?

Question

Let $a = [a_1,a_2,\dots,a_n]^T$, $b = [b_1, b_2, \dots, b_n]^T$.

and $C = \begin{bmatrix}c_1^Tc_1 & c_1^Tc_2 & \dots & c_1^Tc_n\\c_2^Tc_1 & c_2^Tc_2 & \dots & c_2^Tc_n\\ \vdots & \vdots & \ddots & \vdots\\c_n^Tc_1 & c_n^Tc_2 & \dots & c_n^Tc_n\end{bmatrix}$

Where ($c_i$) is a list of vectors of the same size.

In order to get $a_i = b_i - \sum_{j = 1}^na_jc_j^Tc_i$

Should it be $a = b - C^Ta$ ? In the book which I am reading, they write $a = b - Ca$.

Please just give me a confirmation, am I correct?

Answer 1

C is $n\times n$ matrix and $C^T=C$ so $Ca=C^Ta$.