Can all known ML algorithms be written as a sequence of matrix operations?
Linear algebra is very important for ML, but ML is not "just" linear algebra.

Any sequence of linear matrix-vector operations is linear.
Therefore any non-linear operation cannot be computed using linear algebra. 

A few examples of non-linear operations that are used in ML:


The if statement: e.g. (x &lt; 0) 
The softmax activation
The rectified linear unit



  Can advanced features like convolution be expressed as basic matrix
  operations?


This statement is true actually. A single convolution can be expressed as matrix. However, any modern CNN has non-linearities following the convolution.