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In Least squares optimization,
I have $A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 1 & t_n & t_n^2 & \cdots & t_n^k \end{pmatrix}$
and b=\begin{pmatrix} s_1 \\ s_2 \\ \vdots\\ s_n\\ \end{pmatrix} c=b=\begin{pmatrix} x_0 \\ x_1 \\ \vdots\\ x_n\\ \end{pmatrix}

And $f(x_0,x_1,...,x_n)=\left\lVert b-Ax\right\rVert ^2=(b-Ax).(b-Ax)=b.b-2b.Ax+Ax.Ax$.
I understand up to this but I don't understand in the next step how they change $b.Ax$ to $A^Tb.x$ and $Ax.Ax$ to $x.A^TAx$ .
That is I don't understand the step $b.b-2A^Tb.x+x.A^TAx$
Is there a rule on this dot products and matrix transpose?

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  • $\begingroup$ It's enough to note that for any two vectors, $x.y = x^Ty$. $\endgroup$ – Omnomnomnom Dec 8 '15 at 16:06
  • $\begingroup$ @Omnomnomnom Then shouldn't $b.Ax$ be $b^TAx$ $\endgroup$ – clarkson Dec 8 '15 at 17:44
  • $\begingroup$ Right, and that's the same as $(A^Tb)^Tx$ $\endgroup$ – Omnomnomnom Dec 8 '15 at 18:45

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