Let $f_k$, $k=1,2,\dots n$ are linearly dependent continuous functions on $(-1,1)$. Then $\int_0^x$ $f_k(t)dt,$ $k=1,2, \dots, n$ are linearly dependent functions on $(-1,1)$.
Would this be true or false?
See this is what I am confused about, mainly where to start, and how I can apply Linear Dependency with the problem. Maybe I truly don't understand Linear Independence/Dependence.
On
If the functions are linearly dependent, you know that at least one of them,say $f_1$ can be expressed in terms of the others, $$ f_1(x)=\sum_{k=2}^n c_k\,f_k(x)\;.$$ But then $$ \int_0^xf_1(t)\,\mathrm{d} t=\sum_{k=2}^nc_k\,\int _0^xf_k(t)\,\mathrm{d}t\;,$$ showing that the functions $F_i(x)=\int_0^xf_i(t)\,\mathrm{d}t$ are linearly dependent.
The answer is yes. Since $f_i,i=1,\dots,n$ are linear dependent, there exist $\alpha_i,i=1,\dots,n$ such that $$ \sum_{i=1}^n \alpha_i^2 \neq 0 \textrm{ and} \sum_{i=1}^n \alpha_i f_i(x)=0,\forall x\in(-1,1) $$ Then integrate above equation from -1 to x we can get $$ \sum_{i=1}^n \alpha_i \int_{-1}^x f_i(t)dt=0,\forall x\in(-1,1) $$ It follows that $\int_{-1}^x f_i(t)dt,i=1,\dots,n$ are linear dependent.