2: Linear Transformations and Matrix Algebra
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- 2.1: Matrix Transformations
- In this section, we learn to understand matrices geometrically as functions, or transformations. We briefly discuss the types of transformations that matrices can induce, and then discuss special types of transformations that come from matrices.
- 2.2: Linear Transformations
- In this section, we ask two important questions: (1) How can we tell if a transformation is a matrix transformation? (2) If our transformation is a matrix transformation, how do we find its matrix?
- 2.3: Matrix Algebra
- In this section, we study the algebra of transformations and use this to define the algebra of matrices, including addition, scalar multiplication, composition, and inverses.
- 2.4: Invertibility
- In this section, we learn when it is possible to find an inverse of a given linear transformation or matrix.
- 2.5: Determinants- Definition
- In this section, we define the determinant, and we present one way to compute it. Then we discuss some of the many wonderful properties the determinant enjoys.
- 2.6: Cofactor Expansions
- In this section, we give a recursive formula for the determinant of a matrix, called a cofactor expansion. The formula is recursive in that we will compute the determinant of an n×n matrix assuming we already know how to compute the determinant of an (n−1)×(n−1) matrix. At the end is a supplementary subsection on Cramer’s rule and a cofactor formula for the inverse of a matrix.
- 2.7: Determinants and Volumes
- In this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining properties of the determinant. It is also a crucial ingredient in the change-of-variables formula in multivariable calculus.