Can someone clarify what the linear assumption of PCA is?
PCA is the best (in the mean-squared error sense) linear decomposition method.

PCA is defined as an orthogonal linear transformation that transforms
the data to a new coordinate system such that the greatest variance by
some scalar projection of the data comes to lie on the first
coordinate (called the first principal component), the second greatest
variance on the second coordinate, and so on.
Wikipedia

The term &quot;Linear&quot; in PCA means:
a. That any data point is simply a linear combination of the principal components.
b. That the data matrix ($A$) can be decomposed via linear similarity transformations to diagonal matrix ($\Sigma$).
Ie
$$A = U \Sigma U^T$$
or
$$AU = U \Sigma$$
$\Sigma$ is the diagonal matrix of variances for each basis vector.
One can see at once that the linear algebra of the above formula makes clear the meaning of Linearity in PCA.
On the other hand decomposition methods like ICA (Independent Component Analysis) cannot be expressed via linear algebra as PCA above, since they require not only decorrelated components but independent components which is a stronger condition requiring non-linearities.
See also: https://datascience.stackexchange.com/a/80361/100269