1: Systems of Linear Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
- 1.1: Systems of Linear Equations
- In this section, we discuss the geometric shapes of points, lines, and planes and the domains they sit in: Rn
- 1.2: Row Reduction
- In this section, we will present an algorithm for “solving” a system of linear equations.
- 1.3: Vectors
- In this section, we define the concept of vectors in Rn and their basic algebra, as well as the concepts of linear combinations and spans.
- 1.4: Matrix Equations
- In this section, we introduce a very concise way of writing a system of linear equations using matrices and vectors: A\vec{x}=\vec{b}. We also explore the reason our solution sets come in particular shapes for these equations.
- 1.5: Linear Independence
- In this present section, we formalize this idea in the notion of linear independence in order to write spans of vectors efficiently.
- 1.6: Subspaces
- In this section, we describe the concept of a subspace of Rn as a collection of vectors which is closed under addition and scalar multiplication, as well as some relationships between matrices and certain subspaces.
- 1.7: Basis and Dimension
- In this section, we define a basis of a subspace. We then use this to find relationships between the "size" of the null space for a given matrix (the size of a solution set to A→x=→b when it exists), and the "amount" of vectors →b which make such an equation consistent.