I have the following question here.
Define functions $f_1,f_2,f_3 \in F$ by $$f_1(x)=1,f_2(x)=\cos(x),f_3(x)=\cos\left(x+\frac{\pi}{4}\right)$$ and let $V=\text{Span}(f_1,f_2,f_3)$. Are the spaces $\mathbb{R}^3$ and $V$ isomorphic to each other? If so, provide an isomorphism $\varphi:\mathbb{R}^3 \rightarrow V$. Otherwise, explain why not.
I know the vectors are isomorphic since the dimension of $V$ and $\mathbb{R}^3$ are the same (Since $f_1,f_2,f_3$ are linearly independent, they can't be written in terms of each other so the dimension of $V$ is $3$ as well).
How do I find the actual isomorphism though? We didn't really do many examples of this in class. I know we need to find an invertible linear transformation but I am genuinely stuck on how to do that.
Any help would be greatly appreciated!
Check that whether $f_1,f_2,f_3$ are linearly independent if yes then dimension of $V=3$. Note that If $V$ and $W$ be finite dimensional vector space then $V\cong W.$ You can find this theorem in each book probably. Good luck!