A really simple question. For four $n$-dimensional vextors $a,b,x,y \in \mathbb{R}^n$ and a scalar $C \in \mathbb{R}$, it is known that $b^Ty=a^Tx+C$. Having another vector $d \in \mathbb{R}^n$, how can I get the analytical expression of $d^Ty=f(x)$ in terms of $x$? Or is there any way to verify that such an expression is not possible? Many thanks!
2026-04-13 17:57:09.1776103029
Rewrite expression $b^Ty=a^Tx+C$
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It should not be possible, because of lack of information. Putting a vector $y$ on one side of an equation, means that we need to know an equation for each element of $y$, namely $y_i,i=1,2,...,n$. But the only given information is the inner products, which takes the vectors to a one dimensional space.