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In Exercises $7-12$ , assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems. $$2^{k}, 4^{k},(-5)^{k} ; y_{k+3}-y_{k+2}-22 y_{k+1}+40 y_{k}=0$$

Since by the basis theorem, we have "If a linearly independent set of n-vectors in an n- dimensional space is automatically a basis".

Calculus 3

Chapter 4

Vector Spaces

Section 8

Applications to Difference Equations

Vectors

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So we assume that these three signals So all of these different situation and we need to solve that they form a basis for the solution set off the security firm now by I feel him of 17. We have the dimension off. The solution said off these evacuation is three as it is the order of dedication. Now we present by the castrati test no live to defend him well, that the solutions are linearly independent and we evaluate the casserole automatics. And according to the conservative test, we just need to find one case that's that that the turbulent off these Matt Tricks is non zero, meaning that the mother's is in vegetable. So we proceed by writing down the metrics for K equals zero. That's for simplicity. And we have three. So the power of zero, which is one and then three and then so X squared and then minus five to the power off zero, which is still one and then my miss five and then minus five squared, which is 10 to 5. Hey, no, we do some Lynn around the veterans formations and we just ce obstruct the columns and, uh, find that is the determinant acquires the eminent off this mother tricks with zero here and one here. And, uh, mind most seven here. And, uh, lastly, not this is to square. Leslie would have five in here and, uh, mine on and 21 in here. So these determinant equals the determined these metrics. One timers 75 and 21 wigs, ecu, oz toe into one plus seven, multiplied by five. And, uh, circles 21 class for the plus 35 Hank ones 56 which is positive. And we're done. And now we return toe the base, steal him on base on Bales Tau Tau nine. And we have that These three linearly independent solution seems there are three and they are elements of a vector space. We of that thes any solutions, form form a basis for the solution. Set

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