Looking for integral-free DSP textbooks
I would like to recommend a great book to you (see below), but before that I would like to point out that there is some confusion in your question. If you see a convolution integral in a textbook, then this refers to the convolution of two continuous time signals, which can't be represented by a matrix/vector multiplication. You must be referring to the convolution of two sequences (discrete-time signals), but this convolution is always written as an infinite sum (and not as an integral). For finite length sequences, this sum can indeed be written as a (finite) matrix/vector product.

The book I was referring to is the excellent (and free!) book Foundations of Signal Processing by Vetterli et al. It also stresses (among many other things) the matrix view of linear operators on sequences. I'm especially referring to chapter 3 ('sequences and discrete-time systems'), but I would actually recommend to read the whole book if you want a thorough and modern treatment of signal processing. Disclaimer: the book is not integral-free, but this shouldn't put you off, just give it a try.