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Linear Algebra for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

Steven Clontz, Drew Lewis

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Section 5.4 Eigenvectors and Eigenspaces (GT4)

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Subsection 5.4.1 Warm Up

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Activity 5.4.1.

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Which of the following vectors is an eigenvector for A=[24−1−500−3−91102−2−235]?
  1. [−2101]
  2. [−33−21]
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Subsection 5.4.2 Class Activities

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Activity 5.4.2.

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It’s possible to show that −2 is an eigenvalue for [−14−22−79304].
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Compute the kernel of the transformation with standard matrix
A−(−2)I=[?4−22?930?]
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to find all the eigenvectors x→ such that Ax→=−2x→.
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Definition 5.4.3.

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Since the kernel of a linear map is a subspace of Rn, and the kernel obtained from A−λI contains all the eigenvectors associated with λ, we call this kernel the eigenspace of A associated with λ.
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Activity 5.4.4.

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Find a basis for the eigenspace for the matrix [00310−1013] associated with the eigenvalue 3.
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Activity 5.4.5.

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Find a basis for the eigenspace for the matrix [5−2046−215−212−345−36] associated with the eigenvalue 1.
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Activity 5.4.6.

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Find a basis for the eigenspace for the matrix [4300330000250002] associated with the eigenvalue 2.
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Subsection 5.4.3 Individual Practice

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Activity 5.4.7.

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Suppose that T:R2→R2 is a linear transformation with standard matrix A. Further, suppose that we know that u→=[1−1] and v→=[2−3] are eigenvectors corresponding to eigenvalues 2 and −3 respectively.
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(a)
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Express the vector w→=[21] as a linear combination of u→,v→.
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Subsection 5.4.4 Videos

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Figure 62. Video: Finding eigenvectors
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Exercises 5.4.5 Exercises

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Subsection 5.4.6 Mathematical Writing Explorations

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Exploration 5.4.8.

Given a matrix A, let {v1→,v2→,…,vn→} be the eigenvectors with associated distinct eigenvalues {λ1,λ2,…,λn}. Prove the set of eigenvectors is linearly independent.
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Subsection 5.4.7 Sample Problem and Solution

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Sample problem Example B.1.25.
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