Stephen Boyd and Lieven Vandenberghe (also authors of a well-known book on convex optimization) have recently written an introductory linear algebra book, Introduction to Applied Linear Algebra: Vectors, Matrices and Least Squares that may suit your needs. From the preface:
This book is meant to provide an introduction to vectors, matrices, and least
squares methods, basic topics in applied linear algebra. Our goal is to give the
beginning student, with little or no prior exposure to linear algebra, a good grounding in the basic ideas, as well as an appreciation for how they are used in many
applications, including data fitting, machine learning and artificial intelligence, tomography, navigation, image processing, finance, and automatic control systems.
Also from the preface:
The book covers less mathematics than a typical text on applied linear algebra.
We use only one theoretical concept from linear algebra, linear independence, and
only one computational tool, the QR factorization; our approach to most applications relies on only one method, least squares (or some extension). In this sense
we aim for intellectual economy: With just a few basic mathematical ideas, concepts, and methods, we cover many applications. The mathematics we do present,
however, is complete, in that we carefully justify every mathematical statement.
In contrast to most introductory linear algebra texts, however, we describe many
applications, including some that are typically considered advanced topics, like
document classification, control, state estimation, and portfolio optimization.
The book is indeed replete with applications, many of which come from machine learning (or data mining) and from statistics in the guise of data fitting. It's very much an introduction though, as you'll need another source for learning topics such as vector spaces, subspaces, LU decomposition, eigenvalues, eigenvectors, the singular value decomposition, and various other topics. It's very much at the other end of the spectrum from Axler's book, but they do have something in common - little or no emphasis on determinants. In fact, the word "determinant" does not even appear in Boyd & Vandenberghe.
On balance, I think it's worth a look as a relatively gentle introduction with a wealth of applications.