MAT4858 Assignment 1 (197343) Due 30 June 2026 |Matrix Theory and Linear Algebra II|

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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MAT4858: Linear Algebra

Assignment 01 — Semester 1, 2026

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MAT4858
Module Code:
Linear Algebra
Module Name:
Assignment 01
Assignment Number:
197343
Unique Number:
30 June 2026
Due Date:
50
Total Marks:




Submitted in partial fulfilment of the requirements for MAT4858 — UNISA 2026

, UNISA | MAT4858 Assignment 01 — 2026



Question 1: Eigenvalues and Eigenvectors (10 Marks)


1.1 Eigenvectors and the Same Eigenvalue


Question: Let α be an endomorphism of a vector space V over a field F and let v and w be
eigenvectors of α. If v + w ̸= 0V , show that v + w is an eigenvector of α if and only if both v
and w correspond to the same eigenvalue.


The proof proceeds by establishing both directions of the biconditional.


Part A: Assume v + w is an eigenvector of α, and show λ = µ


Since v is an eigenvector, there exists a scalar λ ∈ F such that


α(v) = λv.



Since w is an eigenvector, there exists a scalar µ ∈ F such that


α(w) = µw.



Since v + w is an eigenvector by assumption, there exists a scalar γ ∈ F such that


α(v + w) = γ(v + w).



By the linearity of α,
α(v + w) = α(v) + α(w).


Substituting the eigenvalue equations,


λv + µw = γ(v + w).



Expanding the right-hand side,
λv + µw = γv + γw.




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