## matrices as an array of numbers

At present I introduce matrices as an array of numbers and then carry out various matrix operations. Is there a more tangible way of introducing this topic?
I have thought of transformations but my experience with students has been that they get lost in the transformations and so give up on matrices.
These are weak engineering students who struggle and think of mathematics as too abtract.

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 How about describing them as (special kinds of) functions? Engineers certainly know functions and feel familiar with them. They certainly also understand that functions might be important.
 What?! An amazing attitude sir! Calling the students weak before you have even broached the topic? I hope you are not at a publicly funded institution! Go through Gilbert Strang's MIT OCW videos on youtube, try to see how he motivates the topic. He does not do a very good job. Then instead of portraying arrays as being 'collection of numbers' show them what they can do visually. Look at the CS applications of linear algebra for this. They normally present images of rotations and scaling and other transformations of vectors by linear algebra operations. Then imagine questions like, 'given a linear array of letters, how can I find nested combinations of letters in there? eg. asdflkasdfljasdfjadadfadffgadfasdfa, how can you automatically find and and all the letters in between?' Well, this particular example is esoteric, but surely you, strong teacher, have enough creativity to find questions possessing such simple characteristics? Or how about, given a black/white image, essentially an MxN array, find out how many connected components are there? Or how about the basics of graph theory? I could go on with CS apps here. But let's get back to engineering. Linear algebra is used in the state space representation of ODEs in controls class, coupled system vibration analysis (similar concept), and solid mechanics (principle stress components of a stress tensor, the invariance concept can be introduced here) and solutions to finite element/difference system of equations, fracture mechanics (Williams solution for stress singularity is based on the solution of a homogeneous linear system), linear optimization, think about the Jacobian or hessian matrices, you extract the eigenvalues there to recognize system singularities. It's easy to go on here. You would do well to introspect about your own learning before preaching in public. Every engineer understands ODEs. Show them how the characteristic equation to the homogeneous equation obtained from 'assuming' the solution to be $\text{e}^{rx}$ and the very equation itself is the same as the characteristic equation and eigenvalues obtained from the eigenanalysis of a matrix. Show them how the linear algebra concept of linear independence applies in this very case as well and how you employ that concept in formulating the general solution. Pardon my hostility, but I do not think highly of instructors who consider their students weak. They only succeed in conveying their personal sense of confusion which results from their own ennui, in class. It is but a bad teacher who calls his acolytes that. You do a shoddy job now, and your students end up suffering for the next few years of their undergraduate as this is one of the most fundamental topics in engineering. Your job is not writing grant proposals! Your job is to teach them, get them interested if they are not; not molest them into a state of intellectual submission and scar them forever! In other words don't screw with who gives you your basic pay!!

 Tags engineering, linear algebra, mathematics, matrices, transformations