>>14131191There are three levels to mathematics. The first is carrying out the operations to solve a problem. The second is understanding the intuition of why those operations work. The third is manipulating - or even creating - systems of formal laws.
Given linear algebra the first level let's you find the solutions to a linear system or define what linear dependence is. This is what schools are concerned about, if you have this they will give you full marks. At the 2nd deeper level it's about why elementary row operations work. They work because you're actually replacing the lines with other lines that intersect at the same point, and that testing for linear dependence is equal to finding if the lines if intersecting at zero are the same line or different. That intuitive understanding is a framework that let's you understand more things, like how linearly independent lines can span the entire Cartesian space if they equal the number of dimensions. The third level is when you can use these to build your own systems at a very high based on these rules, like image recognition.
