How to prove that the linear independency of $f_{i}$ from $C^{n-1}\left [ a, b \right ]$ to $C\left [ a, b \right ]$ How to prove that > if $f_{1}, f_{2}, ..., f_{n}$ are linearly independent in $C^{n-1}\left [ a, b \right ]$, > > then they will also be linearly independent in $C\left [ a, b \right ]$.

The inclusion $$I:C^{n-1}[a,b]\to C[a,b]$$ tautologically defined by $I(f)=f$ is a linear map which is injective. Therefore if $$C[a,b]\
i 0=\sum\lambda_i f_i=\sum\lambda_iI(f_i)=I(\sum\lambda_if_i)$$

we must have $\sum\lambda_if_i=0\in C^{n-1}[a,b]$. If $f_i$ are linearly independent on $C^{n-1}[a,b]$ then $\lambda_i=0\forall i$, so $f_i$ are independent as vectors of $C[a,b]$

This works in general if you replace $I:C^{n-1}[a,b]\to C[a,b]$ whith any linear injection $f:V\to W$ between vector spaces.