A basis $B=[b_1\ \cdots\ b_d]$ contains independent vectors spanning $\mathbb{R}^d$. Coordinate vector $c$ represents geometric vector $x$ through $x=Bc$. Because $B$ is invertible, $c=B^{-1}x$. The object has not changed; only the numbers used to describe it have changed.
Let linear map have standard-coordinate matrix $A$, so $y=Ax$. Writing $x=Bc$ and $y=Bc'$ gives $$c'=B^{-1}ABc.$$ Therefore the same map in the new basis has matrix $A'=B^{-1}AB$. This similarity transformation preserves intrinsic properties such as determinant, trace, and characteristic roots.
An invariant direction $v\neq0$ satisfies $Av=\lambda v$. Along that direction, the map changes only scale, and possibly sign, by factor $\lambda$. If a basis can be formed from independent invariant directions, place them in $B$ and obtain $B^{-1}AB=\Lambda$, where $\Lambda$ is diagonal.
Diagonal form makes repeated application simple: $A^k=B\Lambda^kB^{-1}$. Each coordinate evolves independently as $\lambda_i^k$. This explains growth, decay, and oscillatory sign changes in linear dynamics. A magnitude greater than one grows, less than one decays, and equal to one preserves magnitude along that direction.
Not every matrix has enough independent invariant directions for diagonalization. Even when it does, basis matrix $B$ may be poorly conditioned, so small input or rounding errors become large coordinate errors. Symmetric real matrices are especially well behaved because they admit an orthonormal invariant basis with $B^{-1}=B^T$.
In machine learning, basis changes appear in feature whitening, spectral analysis, and preconditioning. Always distinguish a coordinate transformation from changing the underlying function. Verify reconstruction $x\approx B(B^{-1}x)$, monitor the condition number of $B$, and avoid explicit matrix inversion when a stable linear solve is available.