The thing about linear algebra is that it isn't really a cohesive subject in the same way that calc is. This is why you get as many perspectives as you get commenters.
There are some central concepts, and which ones matter to you depends on what you're planning to do with your future. The key points are:
* Matrices/vectors as grids of numbers/computational tools.
* Matrices/vectors as positions and transformations of those positions.
* Matrices/vectors as more abstract geometric objects
* Matrices/vectors as algebraic objects.
There is no 'done right' imo. My sense is LADW is probably the best option for a motivated honors math student, because unlike Axler the author doesn't hate determinants for whatever reason. Strang would work well for engineers. Axler is mostly concerned with the last two, but IMO this makes him kinda niche.
For math majors, LADR is far better than LADW, almost most of LA books use the determinant to prove the advanced stuff (diagonlaization etc), what sets LADR apart is the intentional delay of introducing determinants in order to prove (and hence understand) what's the goal behind concepts such as diagonalization, I love this method.
honestly students should just study ode's to motivate and learn linear algebra which otherwise is lacking substance, and peek into pde's to see why vector space necessary and insightful especially for nonlinear pde.
There are some central concepts, and which ones matter to you depends on what you're planning to do with your future. The key points are:
* Matrices/vectors as grids of numbers/computational tools.
* Matrices/vectors as positions and transformations of those positions.
* Matrices/vectors as more abstract geometric objects
* Matrices/vectors as algebraic objects.
There is no 'done right' imo. My sense is LADW is probably the best option for a motivated honors math student, because unlike Axler the author doesn't hate determinants for whatever reason. Strang would work well for engineers. Axler is mostly concerned with the last two, but IMO this makes him kinda niche.