From what I understand from the question, the matrix ## f_{i}(x_{j}) ## is a ## 3 x 3 ## matrix whose rows are linearly independent.
That means that we are given that ## \alpha f_{1} + \beta f_{2} + \gamma f_{3} = 0 ## implies that ## \alpha = \beta = \gamma = 0 ##.
I believe I have to show that...
There's a question in charles curtis linear algebra book which states:
Let ##f1, f2, f3## be functions in ##\mathscr{F}(R)##.
a. For a set of real numbers ##x_{1},x_{2},x_{3}##, let ##(f_{i}(x_{j}))## be the ##3-by-3## matrix
whose (i,j) entry is ##(f_{i}(x_{j}))##, for ##1\leq i,j \leq 3##...