3 v is a vector in Rn S is the set of all n n matrices for

3. v is a vector in Rn. S is the set of all n × n matrices for which v is an eigenvector. Is S a subspace of Rn×n? Prove your answer

1. Check for closure under addition

Let, P and Q be in this set

So, Pv=pv,Qv=qv for some p,q

(P+Q)v=Pv+Qv=pv+qv=(p+q)v

HEnce closed under addition

2. Check for closure under scalar multiplication

Let, P be in the set and c be a scalar

Pv=tv

(cP)v=cPv=ctv=(ct)v

So, set is closed under scalar multiplication and hence S is a subspace