Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Show that the signals in Exercises $3-6$ form a basis for the solution set of the accompanying difference equation. $$(-3)^{k}, k(-3)^{k} ; y_{k+2}+6 y_{k+1}+9 y_{k}=0$$

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

Johns Hopkins University

Oregon State University

Harvey Mudd College

Boston College

Lectures

02:56

In mathematics, a vector (…

06:36

02:31

Show that the signals in E…

02:12

Verify that the signals in…

01:50

In Exercises $7-12$ , assu…

04:20

01:55

03:38

04:15

00:53

04:25

02:40

in this video, we have a second order difference equation that's provided here. And our goal is to find a basis for H where h is a subspace of this space of all signals. That is the solution to this difference equation. To start out, we have two solutions that we're going to verify. We have for the 1st 1 that y que is equal to negative three to the power of K. What's substitute this solution into the difference equation? To verify that it is indeed a solution. First, why K plus two would be negative three to the power of K plus two plus six times why K plus one is negative three to the power of K plus one plus nine times y que is nine times negative three to the power of K. Now, for this expression we have so far we can factor out negative three to the power of K And then what we're left with is negative three squared plus six times negative three to the power one plus night. It's all together. This is going to result in negative three to the Power k. This will be multiplying the group where we have negative three squared for a 96 by negative three for a negative 18 plus nine, and this results in zero altogether. So we have one solution so far, and keep in mind that are difference. Equation is of second order. That implies that subspace h is of dimension to so if we can find another solution, will be ready to describe the basis for age. So let's consider next y que equals k to the K times Negative three to the power of K For this particular signal. If we substituted into the difference equation here we obtain the following. Why K Plus Two is going to be K plus two times negative three to the power of K plus two. Then we have plus six times why K plus one which is going to be a K plus one times negative three to the power of K plus one. Then lastly, we have plus nine times y que where y que is given to be K times Negative three to the Power K. If this equals zero, then we'll have a second solution and will be ready to describe the basis. Let's use the same strategy as before and factor out negative three to the Power K. This time we're going to be left with K plus two times negative three squared, plus six times K plus one times negative three to the power of one. I'll just write as negative three, then plus nine times K. So let's simplify inside the group first to see what we're dealing with. We have first from here ah value of nine which will distribute into this group giving us nine K plus 18. Next. Considered the six and the negative three. Multiplying them together gives us negative 18 which will distribute into this group that gives us naked of 18 K minus a team. Then copy down Plus nine K. Now all together we have, Let's see here two copies of nine K with negative 18 K that cancels out than 18 minus 18 produces zero. So this entire quantity is euro, and we've just verified that the signal is a solution. And we previously verified that the other signal isas Well, now we're ready to state our conclusion. Let's capture the fact that we have a second order difference equation From that we can say since the dimension of age is equal to two. And since let's use negative three to the power K notation and que times negative three to the power of K are not multiples of each other. We have that negative three of the power of K and K times negative Three of the power of K is a basis four h so sometimes refer to these two particular symbols signals as to be a fundamental set of solutions to this difference equation that we started with. Now that we have a basis, our solution is complete.

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

Show that the signals in Exercises $3-6$ form a basis for the solution set o…

Verify that the signals in Exercises 1 and 2 are solutions of the accompanyi…

In Exercises $7-12$ , assume the signals listed are solutions of the given d…

02:48

Suppose $A$ is $m \times n$ and $\mathbf{b}$ is in $\mathbb{R}^{m} .$ What h…

01:10

Is $\left[\begin{array}{l}{1} \\ {4}\end{array}\right]$ an eigenvector of $\…

06:14

Diagonalize the matrices in Exercises $7-20,$ if possible. The eigenvalues f…

03:26

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has…

03:22

$A$ is a $4 \times 4$ matrix with three eigenvalues. One eigenspace is one-d…

01:13

In the vector space of all real-valued functions, find a basis for the subsp…

05:52

In Exercises $25-28$ , show that the given signal is a solution of the diffe…

07:27

In Exercises $13-16,$ define $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$…

01:54

Construct an example of a $2 \times 2$ matrix with only one distinct eigenva…

01:18

Find a basis for the set of vectors in $\mathbb{R}^{2}$ on the line $y=5 x$<…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.