MATH 121A, Winter 2026 Discussion Sect. 5-1

MATH 121A Discussion Section 5-1

10 Mar 2026

1 Section 5-1. Eigenvalues and Eigenvectors

1.1 Worked Examples

Example 1. Let V be a vector space and T : V → V be linear. Suppose T 2 - 3T + 2IV = T0.

Prove that if λ ∈ F is an eigenvalue of T, then λ = 1 or 2.

Solution.

Let λ be an eigenvalue of T and x be an eigenvector corresponding to the eigenvalue λ. Then

T2(x) = T(T(x)) = T(λx) = λ(T(x)) = λ(λx) = λ2x.

so

0 = T0(x) = (T2 - 3T + 2IV )(x) = (λ2 - 3λ + 2)x

since x ̸= 0, we have λ2 - 3λ + 2 = 0, hence λ = 1 or 2.

Example 2. Let V be a vector space and T : V → V be linear. Suppose there are non-zero

vectors u, w ∈ V such that

T(u) = 3w and T(w) = 3u.

Prove that 3 or -3 is an eigenvalue of T.

Solution.

Since T2(u) = T(3w) = 3T(w) = 3(3u) = 9u and u ̸= 0, u is an eigenvector of T 2 with

eigenvalue 9, hence T2 - 9IV is not 1 - 1. So (T - 3IV )(T + 3IV ) is not 1 - 1. Hence T - 3IV

or T + 3IV is not 1 - 1. This is equivalent to say 3 or -3 is an eigenvalue of T.