My answer to a question on Quora:
Mathematics is the art of avoiding calculations, but to avoid calculations, you have to understand their working. It is not clear from your question whether you reached the level of understanding of row operations which allows you to move further. Solution of each of the following problem involves Gaussian elimination.
- Can you compute, using row operations, say, a
determinant of size 4 by 4 or 5 by 5 ? For simplicity,
assume in this and following problems that all coefficients are single digit integers.
- Invert a 4 by 4 or 5 by 5 matrix matrix?
- Do you understand that row operations can be obtained by multiplying by elementary matrices, and can you find these matrices?
- Can you find a reduced echelon form of, say, 3 by 6 matrix and use it to read the coefficients of linear dependencies between columns of the original matrix?
- Do you understand why the reduced echelon form of a matrix is unique?
- Given a subspace spanned by some vectors in the vector space, can you find a basis of this subspace?
- Can you find a basis of the intersection of two vector subspaces given by their generating sets?
- Can you continue this list of problems of linear algebras which are solved with the help of Gaussian elimination?
This is the Catch-22 of learning mathematics: only at the next stage of learning it becomes possible to tell whether the learner mastered the previous stage. Unfortunately, school and university curricula do not have sufficient slack, free space, to allow students to come back and revisit the previous material every time when this is needed. Learning mathematics is an ascending spiral; schools and universities do their best to straighten and flatten it, and in the process break it in pieces.
My advice: if you think you understand Gaussian elimination, try to move further. Work on bullet points above will involve Gaussian elimination, and will train your skills in Gaussian elimination, but, hopefully, will be less boring than elimination for the sake of elimination (which is boring indeed). But as soon as your progress grinds to halt, be prepared to return back to basics and learn to do Gaussian elimination without “silly mistakes”.
And even more efficient remedy: this is perhaps time-consuming, but learn to code in some computer language and write at least some bits of linear algebra that you learn, as computer codes / programmes without using built-in functions which could be already present in the language – and make sure that you codes work.
My next piece of advice is more delicate. I apologise for its excessively personal nature. An attempt at code writing – if it fails – might expose a deeper problem. Do you have a sufficient attention span? This is a mental trait that is much needed in mathematics, but no longer required or supported by everyday life of modern society. If you feel that you make “silly mistakes” because of lapses in attention (which you may feel as “boredom”), try needlework, or Meccano, or jigsaws. Of the latter, I would recommend a jigsaw of the classical London Underground map – it is mathematics. I know that, I did this jigsaw with my grandson, when he was 6 years old; at some point we agreed to ignore the boring white bits – recovery of the structure was much more interesting. In Linear Algebra, there is a plenty of “boring white bits”; the trick is to tell them apart from the critical structural bits.
I checked: it is available on Amazon.