# Help with Linear Algebra: Find Solutions & Verify

• shan
In summary: Thanks for your help!In summary, the conversation revolved around two different questions. The first one involved finding an expression for x_n in a given matrix equation, which required an eigenvector decomposition. The second question involved verifying that there are infinitely many least squares solutions for a given system of equations, which was solved by multiplying both sides by the inverse of the matrix and then row reducing. The conversation also included some clarifications and corrections regarding the solutions provided.
shan
This first one, I just want to verify that I've understood what the question is asking.

It says: If $$x_{n+1} = Ax_{n}$$, write an expression for $$x_{n}$$. The matrix A = $$\left(\begin{array}{ccc}2&0&0\\1&3&0\\-3&5&4\end{array}\right)$$

From what I understand, this question wants me to do an eigenvector decomposition so this is what I came up with (after finding the eigenvectors and eigenvalues):
$$x_{n} = c_{1}(4 * \left(\begin{array}{c}0\\0\\1\end{array}\right)) + c_{2}(3 * \left(\begin{array}{c}0\\\frac{-1}{5}\\1\end{array}\right)) + c_{3}(2 * \left(\begin{array}{c}\frac{1}{4}\\\frac{-1}{4}\\1\end{array}\right))$$
Is that what the question was asking for??

The other question (or help I need) is this one: Verify that there are infinitely many least squares solutions which are given by x = $$\left(\begin{array}{c}\frac{2}{7}\\\frac{13}{84}\\0\end{array}\right) + t \left(\begin{array}{c}\frac{-1}{7}\\\frac{5}{7}\\1\end{array}\right)$$
$$\left(\begin{array}{ccc}3&1&1\\2&-4&10\\-1&3&-7\end{array}\right) \left(\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}\right) = \left(\begin{array}{c}2\\-2\\1\end{array}\right)$$

I multiplied both sides by the inverse of the matrix and got this system:
$$\left(\begin{array}{ccc}14&-8&30\\-8&26&-60\\30&-60&150\end{array}\right) \left(\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}\right) = \left(\begin{array}{c}-1\\13\\-25\end{array}\right)$$
and I then row reduced it to..
$$\left(\begin{array}{cccc}14&-8&30&1\\0&\frac{150}{7}&\frac{-300}{7}&\frac{95}{7}\\0&0&0&0\end{array}\right)$$
I thought I was going on the right track since I can see that this has infinitely many solutions but when I tried to find x...
$$x_{3} = t$$
$$x_{2} = \frac{19}{30} + 2t$$
$$x_{1} = \frac{13}{30} - t$$
which definitely does not verify the above question. Can someone tell me where I went wrong? Thanks :)

$$x_{n} = c_{1}(4 * \left(\begin{array}{c}0\\0\\1\end{array}\right)) + c_{2}(3 * \left(\begin{array}{c}0\\\frac{-1}{5}\\1\end{array}\right)) + c_{3}(2 * \left(\begin{array}{c}\frac{1}{4}\\\frac{-1}{4}\\1\end{array}\right))$$

Did you notice that there is no "n" in that? Are you claiming that xn is constant and does not depend on n? There should be some nth powers in there.

In (2) you've completely neglected the "least squares" requirement.

HallsofIvy said:
$$x_{n} = c_{1}(4 * \left(\begin{array}{c}0\\0\\1\end{array}\right)) + c_{2}(3 * \left(\begin{array}{c}0\\\frac{-1}{5}\\1\end{array}\right)) + c_{3}(2 * \left(\begin{array}{c}\frac{1}{4}\\\frac{-1}{4}\\1\end{array}\right))$$

Did you notice that there is no "n" in that? Are you claiming that xn is constant and does not depend on n? There should be some nth powers in there.

Ah, I think I see what you mean... so should it have been more like..
$$x_{n} = 4^(n+1) c_{1} \left(\begin{array}{c}0\\0\\1\end{array}\right) + 3^(n+1) c_{2} \left(\begin{array}{c}0\\\frac{-1}{5}\\1\end{array}\right) + c^(n+1) c_{3} \left(\begin{array}{c}\frac{1}{4}\\\frac{-1}{4}\\1\end{array}\right)$$

In (2) you've completely neglected the "least squares" requirement.
What do you mean by that? I thought that to solve a least squares problem, you use $$A^T Ax = A^T b$$ where x represents the unknowns, in this case, x1, x2 and x3?

Oops, nevermind about the second question, I figured it out :)

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with the study of vectors, vector spaces, linear transformations, and systems of linear equations. It is used to solve problems in various fields such as physics, engineering, computer graphics, and economics.

## 2. How can I find solutions to linear algebra problems?

To find solutions to linear algebra problems, you can use various methods such as Gaussian elimination, matrix inversion, and Cramer's rule. You can also use online calculators or software programs specifically designed for solving linear algebra problems.

## 3. Can linear algebra problems have multiple solutions?

Yes, linear algebra problems can have multiple solutions. This is because a system of linear equations can have either one unique solution, no solution, or infinitely many solutions depending on the number of equations and variables in the system.

## 4. How can I verify if a solution to a linear algebra problem is correct?

To verify a solution to a linear algebra problem, you can substitute the values of the variables in the equations and see if they satisfy all the equations. You can also use matrix operations or graphing to check the solution.

## 5. What are some real-world applications of linear algebra?

Linear algebra has various real-world applications such as image and signal processing, data compression, cryptography, and machine learning. It is also used in designing computer graphics, analyzing financial data, and solving optimization problems in engineering and science.

• Introductory Physics Homework Help
Replies
8
Views
819
• Introductory Physics Homework Help
Replies
13
Views
1K
• Introductory Physics Homework Help
Replies
8
Views
822
• MATLAB, Maple, Mathematica, LaTeX
Replies
3
Views
955
Replies
19
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
850
• Introductory Physics Homework Help
Replies
7
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
706
• Linear and Abstract Algebra
Replies
3
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
940