Matrix is not a convenient way to write linear transformation; it is a good way to reason about linear transformation, because among other things matrix product is function composition. Loose definition and incoherent reasoning are not good motivation for anything let alone a mathematical subject. Bringing matrix analysis as an introduction to linear algebra. For example, there are different ways of writing chain rule for matrices with radically different looking formulas; higher derivatives of vector functions are multilinear forms; convergence in multidimensinal space has too many subtleties for beginning students. Solving systems of linear equation is a perfect way of introducing linear algebra. Not only is it vitally important in math and used everywhere, it is also almost half of linear algebra. Gaussian elimination is the elementary school method formulated as an algorithm and, not coincidentally a sequence of linear transformations culminating in LU decomposition because we can represent a sequence of linear transformation as a series of matrix products. Writing Gaussian elimination in partitioned matrix form, which includes regular scalar manipulation as a special case, results in Schur's complement, and taking inverse of that is a good way to deductively prove the matrix inversion lemma, an important formula in numerical analysis and signal processing and usually taught in advanced courses. Nothing I mentioned is inaccessible to an undergraduate. Linear algebra is an unique mathematical subject because it doesn't require much basics to begin and is very visual, unlike calculus which needs a good foundation in analytic geometry and has many techniques what beginning students call tricks, not to mention mental obstacles like limits. Mathematical intuition comes with practice and experience and mathematical maturity, not dumbing down the topic. I agree motivations should be taught along with any mathematical subjects, but they should be mathematical motivations. Applications are important too, but teaching well and thoroughly theory and application at the same time is impossible and undesirable; different people need different applications which in turn emphasize different aspects of theory; it is just too much.