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## Homework Statement

Suppose that {uk} and {vk} are sequences satisfying uk = Auk-1 k = 1, 2, ... and vk = Avk-1 k = 1, 2,... Show that if u0 = v0 then ui = vi for all values of i.

## Homework Equations

uk --> is u subscript k

u0 --> is u subscript 0

uk-1 --> is u subscript k-1

ui --> is u subscript i

## The Attempt at a Solution

Well so far I have...

uk = A^k(u0)

= A^k(a1u1 + a2u2 +...+anun)

= a1(A^k)u1 + a2(A^k)u2 +...+ an(A^k)un

= a1(lambda1^k)u1 + a2(lambda2^k) +...+ an(lambdan^k)un

But since u0 = v0 A^k(u0) = A^k(v0)

But after there I get uncertain cause I think my next steps would be:

= A^k(a1v1 + a2v2 +...+ anvn)

= a1(A^k)v1 + a2(A^k)v2 +...+ an(A^k)vn

= a1(lambda1^k)v1 + a2(lambda2^k)v2 +...+ an(lambdan^k)vn)

Then conclude uk = vk?

Is this correct?

Thanks! :)