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Help!!!!!

im havin problems tryin to do this question

let A be a 3 x 3 matrix havin 2 eigenvalues, a and b, such that a does not equal b. let v1 and v2 be eigenvectors, both corespondin to the eigenvalue a, such that v1 and v2 are not scalar multiples of each other. let v3 be an eignevector corrsponding to the eigenvalue b. you may assume that{v1,v2,v3} is a linear independent set of vectors in R3

a) prove that every linear combination of v1 and v2 is an eigenvector of A of eigenvalue a.
b) prove that if a vector u an element of R3 is an eigenvector of A of eigenvalue a, then u is a linear combination of v1 and v2.

thanks

Reply 1

Jonny W

(a)
For any scalars p and q,

A(p v1 + q v2) = p A v1 + q A v2 = p a v1 + q a v2 = a(p v1 + q v2)

So (p v1 + q v2) is an eigenvector of A with eigenvalue a.

(b)
Since v1, v2 and v3 are linearly independent, they span R^3. So there are scalars p, q and r such that

u = p v1 + q v2 + r v3

Since u is an eigenvector of A with eigenvalue a,

A(p v1 + q v2 + r v3) = a(p v1 + q v2 + r v3) . . . . . (*)

Because the LHS of (*) equals p a v1 + q a v2 + r b v3,

r b v3 = r a v3
r (a - b) v3 = 0
r = 0 . . . . . since v3 is not zero, and a doesn't equal b

u = p v1 + q v2
u is a linear combination of v1 and v2