That's entirely possible, but in the context of introductory books I think it's fair to assume & limit scope to Strang's "Introduction to Linear Algebra" and Axler's "Linear Algebra Done Right."

I am an applications-oriented person and my inclination was to go directly from a matrix/determinant heavy picture into applications. Strang['s intro text] only. I am extremely glad that someone intercepted me and made me get some practice with abstract vector spaces, operators, and inner product spaces first, using Axler. This practice bailed me out and differentiated me from peers on a number of occasions, so I want to pass down the recommendation.