Let $A'$ denote the standard (coordinate) basis in $\mathbb R^n$ and suppose that $T\colon \mathbb R^n \to \mathbb R^n$ is a linear transformation with matrix $A$ so that $T(x) = Ax$. Further, suppose that $A$ is invertible. Let $B$ be another (non-standard) basis for $\mathbb R^n$, and denote by $A_{(B)}$ the matrix for $T$ with respect to $B$.
a) Prove that $A_{(B)}$ is also an invertible matrix.
b) If $\{x_1,\dots,x_k\}$ is a linearly independent set in $\mathbb R^n$, prove that $$\left\{A_{(B)}[x_1]_{(B)}\dots A_{(B)}[x_k]_{(B)}\right\}$$ is also linearly independent, where $[x]_{(B)}$ denotes the $B$-coordinate vector of $x.$
Ok I think I understand now that A_B = P(A_E)P^-1, where P is the change of coordinates matrix. And since A_E is invertible (it says so in the problem) and obviously P is invertible, then the product PAP^-1 gives an invertible matrix. Is P from standard to B coordinates, and P^-1 is from B back to E? And for part (b), I am lost.
Some hints: Let $E$ be the standard basis of $\mathbb R^n$. Then $A_{(E)} = A$ (Why?)
a) Do you know how you can transform matrix $A_{(E)}$ to matrix $A_{(B)}$? Do you know how to characterize invertibility via determinants?
b) How does the linear mapping $f \colon x_i \mapsto [x_i]_{(B)}$ look like (sometimes it's called coordinate transformation)? Is it invertible? What do you know about images of linearly independent vectors under invertible linear mappings? (indeed, injectivity will suffice for this last question)