TBIL-LA Geometric Properties of Linear Maps (GT)

Chapter 5 Geometric Properties of Linear Maps (GT)

How do we understand linear maps geometrically?

By the end of this chapter, you should be able to...

Describe how a row operation affects the determinant of a matrix.
Compute the determinant of a $4 \times 4$ matrix.
Find the eigenvalues of a $2 \times 2$ matrix.
Find a basis for the eigenspace of a $4 \times 4$ matrix associated with a given eigenvalue.

Before beginning this chapter, you should be able to...

Calculate the area of a parallelogram.
- Review: Khan Academy
  ¹
  www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-geometry-topic/cc-6th-parallelogram-area/v/intuition-for-area-of-a-parallelogram
Recall and use the definition of a linear transformation.
- Review: Section 3.1
Find the matrix corresponding to a linear transformation of Euclidean spaces.
- Review: Section 3.2
Find all roots of quadratic polynomials (including complex ones).
- Review: Khan Academy
  ²
  www.khanacademy.org/math/algebra-home/alg-polynomials/alg-factoring-polynomials-quadratic-forms/v/factoring-trinomials-by-grouping-5
  , YouTube (1)
  ³
  youtu.be/Aa-v1EK7DR4
  , YouTube (2)
  ⁴
  www.youtube.com/watch?v=2yBhDsNE0w
Interpret the statement “ $A$ is an invertible matrix” in many equivalent ways in different contexts.
- Review: Section 4.3