Let $n\in \mathbb{N}$ and suppose $X$ is a set with $n$ elements. Define $V$ as the set of all functions $f:X\to \mathbb{C}$. It can be shown that the set $V$ under the usual function $+$ and scalar $\cdot$ is a vector space. We are asked to find a basis for this vector space.
My answer to the problem which I am unsure is:
$B=\{f_1(x)=1, f_2(x)=i\}$ serves a basis for $V$.
It is easy to see that both the given functions in the set $B$ are elements of $V$ and are linearly independent.
However, I had a hard time figuring out that the set of linear combination
$a_1f_1+a_2f_2=\mathbb{C}$ will cover all the elements in $V$.
What I am thinking is that the set of linear combination generates the whole $\mathbb{C}$ and not $V$.
Is my solution correct? Also please help me understand the linear combination part. Thank you so much in advance.
Best Answer
Let $X=\{x_i\}_{i=1}^n$. We can represent any $f\in V$ by $f=(\lambda_i)_{i=1}^n$, where $\lambda_i\in \mathbb C$. This indicates to us that $V$ is of dimension $n$ over $\mathbb C$, or dimension $2n$ over $\mathbb R$, as we are basically considering $\mathbb C^n$. So identify a basis for $\mathbb C^n$, and then translate this back into function notation if you so wish. I.e $(1,0,\dots,0)$ is a basis vector for $\mathbb C^n$, so we can associate with this the function $f_1$, defined by $f_1(x_1)=1$ and $f_1(x_i)=0$ for all $i\neq 1$.