Ashley H.

Let A= a b
c  d   be an arbitrary 2x2 matrix. Let x⊆ Rand suppose that (xn)→x for some x∈R2. Prove that (Axn)→Ax

By:

Ashley H.

Thanks very much :) but we were also just given the following hints, maybe I have to include something relating to them in my answer. Is there a way to include some? Many thanks

1. You may use the fact that (xn) → x implies (x
(n)
i
) → xi
, i = 1, 2 without proof.
2. Realise that Axn − Ax = A(xn − x).
3. It is easier to estimate kA(xn − x)k1 than kA(xn − x)k2. Look at the cover sheet for
a relation between the two.
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02/21/18

Bobosharif S.

tutor
1) You don't have to prove that xni → xi, because it is an assumption here and indeed you are using this fact,
2) Yes, indeed it is better to use Axn − Ax = A(xn − x). As xn → x, A(xn − x)→ 0, which is in principle the same as "by coordinate convergent"
3) I'm not sure what you mean by k, k1 and k2.

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02/21/18

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