Linear Algebra is the branch of mathematics aimed at solving systems of linear equations with a ﬁnite number of unknowns.
- The similarities and differences between RR and CC can be described as elegant and intriguing, but why are complex numbers important? One possible answer to this question is the Fundamental Theorem of Algebra. It states that every polynomial equation in one variable with complex coefficients has at least one complex solution. In other words, polynomial equations formed over CC can always be solved over CC .
- You have probably also learned in physics that space-time has dimension four and that string theories are models that can live in ten dimensions. In this chapter we will give a mathematical definition of the dimension of a vector space. For this we will first need the notions of linear span, linear independence, and the basis of a vector space.
- There are many operations that can be applied to a square matrix. This chapter is devoted to one particularly important operation called the determinant. In effect, the determinant can be thought of as a single number that is used to check for many of the different properties that a matrix might possess. In order to define the determinant operation, we will ﬁrst need to define permutations.
- In Section 6.6, we saw that linear operators on an n-dimensional vector space are in one-to-one correspondence with \(n \times n\) matrices. This correspondence, however, depends upon the choice of basis for the vector space. In this chapter we address the question of how the matrix for a linear operator changes if we change from one orthonormal basis to another.