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Contents Index Dark Mode Prev Up Next \(\newcommand{\markedPivot}[1]{\boxed{#1}}
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Chapter 5 Geometric Properties of Linear Maps (GT)
In this chapter, we continue our study of linear transformations by studying them from a geometric perspective. The tools we explore in this chapter, namely
determinants and
eigenvectors are incredibly pervasive in mathematics and applications thereof (in fact, it is likely you have already used determinants as a computational tool in other courses prior to fully understanding them).
Motivating Question. How do we understand linear maps geometrically?
Learning Outcomes 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.
Readiness Assurance.
Before beginning this chapter, you should be able to...
Calculate the area of a parallelogram.
Recall and use the definition of a linear transformation.
Find the matrix corresponding to a linear transformation of Euclidean spaces.
Find all roots of quadratic polynomials (including complex ones).
Interpret the statement “
\(A\) is an invertible matrix” in many equivalent ways in different contexts.
