In terms of math lessons, algebra equations are often tricky, just to lose one step in the process due to a simple error will throw off the entire answer. Unlike some forms of mathematics, because there are so many steps in the process the student often finds it difficult to get the correct answers. Of course if students have not learned the steps from the beginning they will never success with their math lessons, algebra equations. Algebra is simply not a subject that can be learned if students do not start from the beginning and make sure they master each step along the way.

Many college teachers are just amazed and quite bewildered to find that their undergrad students have not learned the basics sufficiently to understand the more complex theorems. Many students fail their math lessons, algebra and other mathematical concepts in high school and manage to enter college and university without them, often because they are a mature student. Colleges and universities have had to incorporate high school preparatory classes; just to prepare these students sufficiently for the more advanced math lessons - algebra and then statistics and other forms of forms of higher math necessary for their college programs.

Therefore it is the onus of the math teachers to make sure that math lessons, algebra equations are mastered at these fundamental course levels in order for their students to proceed to the higher levels required of them. Many of these students do not understand the concepts of vectors and vector spaces and so earlier math lessons are required before Euclidean spaces and matrices can be taught. Understanding the underlying concepts from which the theorems are built upon is key to success with the algebraic equations and formulas.

Early on in the math lessons, especially at the college level, algebra concepts would have to include lessons on polynomials in order for students to grasp the properties of linear algebra. From this point the students can begin to understand that on complex vector spaces eigenvalues are certain to exist. At this point, the student can grasp the concept of the upper triangle matrix and how a linear operator on a real vector space has an invariant subspace.

Students are mastering their college level courses by this point and are studying orthonormal bases, the Gram-Schmidt procedure and adjoints. Other theorems useful in college math are the spectral theorem, and more. However, none of these college students will be even understood with other the proper high school courses and prerequisites.