Prove that the set S = {x ∈ Rn |Ax = λx} is a subspace of Rn

Question

Prove that the set S = {x ∈ Rn |Ax = λx} is a subspace of Rn

Answer 1

First, Let x and y be in S. Show that x+y is also in S.

Since x and y are in S, Ax = λx and Ay = λy

So, A(x+y) = Ax + Ay = λx + λy = λ(x + y)

Therefore, x+y is in S.

-----------

Second, Let x be in S and let c be a scalar (i.e., a real number). Show that cx is also in S

A(cx) = c(Ax) = c(λx) = λ(cx)

So, cx is in S.

------------------

Third, 0 is in S since A(0) = λ(0)

Since S is closed under addition and scalar multiplication and since 0 is an element of S, S is a subspace of Rⁿ.