How is a change of basis possible?

Question

A basis set is a set of linearly independent vectors in a space that span that space. If converting from one basis set to another in the same space, you're writing vectors in terms of other vectors, so how is linear independence maintained?

Answer 1

A basis is a maximal set of linearly independent vectors in a vector space V. Now a real vector space V can(will) have infinitely many bases. For example $\mathcal{B} =\left\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}\right\}$ and $\mathcal{C} =\left\{\begin{bmatrix}2\\0\end{bmatrix},\begin{bmatrix}0\\2\end{bmatrix}\right\}$ are both bases for $\mathbb{R}^2$. Note that $\begin{bmatrix}2\\0\end{bmatrix}$= 2$\begin{bmatrix}1\\0\end{bmatrix}$+0$\begin{bmatrix}0\\1\end{bmatrix}$ and $\begin{bmatrix}0\\2\end{bmatrix}$= 0$\begin{bmatrix}1\\0\end{bmatrix}$+2$\begin{bmatrix}0\\1\end{bmatrix}$.

So yes, the $\mathcal{C}$ vectors can be written uniquely in terms of the $\mathcal{B}$ basis, but also the other way around. If we added another vector into $\mathcal{B}$, then linear independence would be lost. A change of basis can be thought of as swapping one maximal linearly independent set for another.