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please help im lost

hint: try to write X as a linear combination of v1 and v2

how do i determine that

ok, first of all have you learned what a linear combination is?

not quite im very confused with this topic

ok i see that

so X=2v1+3v2

excellent! that is right.

OK, so now you are probably thinking... now what? :)

yea

ok i get that

Cool! Now, we get \[2A^{10}v_1 + 3A^{10}v_2\] since scalars can be moved around.

so then we can get 2(\[\lambda ^{10}\]v1+3(\[\lambda _{2^{10}}\]v2

precisely!

i just dont know how to get the final answer

After I've just added to the two vector together after multiplying by the scalars.

That 2^(-9) came from \[2(\frac{1}{2})^{10} = 2 \times 2^{-10} = 2^{-9}\]

but however you want to write it :)

ok i tired that final answer and i got it incorrect

could you take a screenshot maybe of what the input looks like?

or else you could just input the decimal answer... maybe the computer would like that better...

i got it correct thanks so much for your help

no problem, good luck on eigenvectors! :)

therefore
\[\large A^{10}x=(S^{-1}\Lambda S)^{10}x=(S^{-1}\Lambda^{10}S)x \]