Bobosharif S. answered • 02/21/18
Tutor
4.4
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Expert in R with 10 years of statistical analysis of data and ..
xn ⊆ R2. Let xn=(x1n, x2n)→(x1, x2) (meaning by coordinate convergence).
Now, Axn=(ax1n+bx2n, cx1n+cx2n) is also a vector and we need to show that both coordinates converge.
Since x1n and x2n converge and a, b are real numbers, then
ax1n+bx2n →(ax1+bx2). The same with (cx1n+dx2n).
Thus,
Axn=(ax1n+bx2n, cx1n+cx2n)→(ax1+bx2, cx1+dx2)=Ax
Bobosharif S.
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
Ashley H.
(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.
02/21/18