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  • $\begingroup$ I think this depends greatly on the audience of the course. In the US, I've never taught a beginning linear algebra course that has a significant percentage of math majors. The courses are largely populated by engineering students who absolutely need to understand how to do linear algebraic computations, and therefore need to understand matrix algebra. I'll also say that I don't think it's ever a mistake to introduce computational methods alongside theory. $\endgroup$ Commented Oct 2, 2016 at 14:13
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    $\begingroup$ Of course computational techniques are essential. But these need to be explained, especially to non-mathematicians. I know of no book on linear algebra focussing primarlly on matrix calculations that does that. The sorts of questions that arise naturally, such as why matrices must both be mxn in order to be able to add them, yet any mxn matrix can be multiplied by any nxp matrix (in the sutable order), remain unanswered, beyond, perhaps the usual cop-out "it proves to be useful". $\endgroup$ Commented Oct 3, 2016 at 21:43
  • $\begingroup$ I agree that far too often the computations are not explained. Conceptual explanations based on linear transformations can make a huge difference. $\endgroup$ Commented Oct 4, 2016 at 3:07
  • $\begingroup$ In particular, this makes it clear that calculus in R^{n} is really linear algebra applied to infinite dimensional vectors spaces of the form F(X,R), where F(X,R) is the vector space of certain real-valued functions on the subset X of R^{b}, where F stands for continuous, or integrable, or differentiable, or ... The difficulty for computation is that, being infinitely generated, matrices are not available. But exploring the general linear algebra of both simultaneously, shows the intimate link between smooth and discrete systems. $\endgroup$ Commented Oct 5, 2016 at 11:25
  • $\begingroup$ Is this perspective on calculus discussed in Jänichs' book? I think I looked at his book at some point, but maybe not very carefully. $\endgroup$ Commented Oct 6, 2016 at 14:58