Signals and systems book with linear algebra approach
I'm not particularly current, my junior level signals and systems text was


  Lathi, Bhagwal. Signal, Systems, and Controls. Intext, 1973


which had a some linear algebra, primarily because it had a control engineering bent.  The last chapter on mixed continuous-discrete systems was a lot to digest. I have come to appreciate it more over the years.

I used an older version of Chen in graduate school and it is very much from a linear algebra perspective, but is not aimed at the same audience Lathi is.


  Chen, Chi-Tsong. Linear system theory and design. Oxford University
  Press, Inc., 1998.


Kailath is a bit harder than Lathi, 


  Kailath, Thomas. Linear systems. Vol. 156. Englewood Cliffs, NJ:
  Prentice-Hall, 1980.


and Luenberberger is very readable.


  Luenberger, David G. Introduction to dynamic systems: theory, models,
  and applications. Vol. 1. New York: Wiley, 1979.


In signal processing,


  Oppenheim, Alan V., and Ronald W. Schafer. Discrete-time signal processing: Pearson new International Edition. Pearson Higher Ed,
  2013.


Is sort of the "standard" text book on signal processing but light on linear algebra.

Strang is good but assumes you have some linear systems background.


  Strang, Gilbert, and Truong Nguyen. Wavelets and filter banks. SIAM,
  1996.


and his book on Linear Algebra is a very good introduction.

Its hard to recommend a book without knowing your back ground, but If I were to ask the same question you did,  I would go with Chen.