Weekly Math Problem #71

Eigenvectors. This week's problem is related to last week's problem, WMP #70. Check it out, if you haven't. Last week we found the eigenvalues for the matrix. This week, we're gonna find the eigenvectors associated with the eigenvalues. Feel free to refer to WMP #45 to help you with the process. 😁 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Substitution        
        ✔️     How to perform row operations
             

WMP! #71 says...

Happy solving!

Check back on Saturday, January 29th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Here's what I did to find the eigenvectors for each eigenvalue: 

And there you have it! The eigenvectors are not the same. In fact, they are linearly independent. 

▪️ Did you find the eigenvectors?
▪️ Have you ever done matrix algebra before?

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
Up next...WMP! #72. 

Cheers!

The Younge Lady

Weekly Math Problem #70

Eigenvalues. This week, linear algebra came to mind. The last time I did a linear algebra related problem on this blog was in WMP #45. Check it out...it may help you with this week's problem. 👀 *clears throat*. Anyhow, eigenvalues are cool and have lots of applications. For more info, check out the Applications of Eigenvalues and Eigenvectors page from Interactive Mathematics website. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the determinant of a 2x2 matrix
        ✔️     How to solve a quadratic equation
             

WMP! #70 wants us to...

Happy solving!

Check back on Saturday, January 22nd for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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🎶Let's get it started...🎶

The process to finding the eigenvalues of a 2x2 matrix requires working with a polynomial called the characteristic polynomial. Ours will be a quadratic polynomial generated from finding the determinant of (Y - λI2), where I2 is the 2x2 identity matrix.

Now that we have the characteristic polynomial, we need to find it roots. The roots of the characteristic polynomial are the eigenvalues of the matrix. Set the characteristic polynomial equal to zero and solve. Ours is factorable, but if it wasn't, then you'd have the option to use other methods for solving a quadratic equation.

The eigenvalues have been found!

▪️ Hopefully you haven't found the process to finding the eigenvalues of a 2x2 matrix daunting.

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
📗 WMP! #71 is up next. 

Cheers!

The Younge Lady

Weekly Math Problem! #45

Diagonalization. I want to go over some linear algebra this week so, here is a problem to help me with that. I landed on diagonalization of a matrix because, for some reason, it stuck with me. I can't really explain why. Check out my other WMPs that involved working with matrices--WMP! #4 and WMP! #14. I don't have much else to say (due to tiiirrreeeddd-ness) 😩. 

⚠️Many steps involved.⚠️To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to perform row operations
        ✔️     How to find the determinant of a matrix
        ✔️     How to solve a quadratic equation
        ✔️     How to multiply matrices
        ✔️     How to find the inverse of a matrix

All calculations can be done without a calculator but, feel free to use a matrix calculator to check your work. Ready?? 😁🥴 WMP! #45 says...

Happy solving!

Check back on Friday, April 9th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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I'm just gonna go 'head and get into this calculation...computation...situation...

😅

Step 1. Find the Characteristic Polynomial

Step 2. Find the Eigenvalues 

Step 3. Find the Eigenspaces and Eigenvectors 

Step 4. Define Invertible Matrix, F, and Diagonal Matrix, D 

Step 5. Complete the Diagonalization

 

(Check the Diagonalization)

Step 6. Complete the Diagonalization

(Final Check)

**This plot was created using Desmos' Matrix Calculator.

WHEW!!! That was a lot. (You can't say I didn't warn you. With that...I think I'm gonna take it easy next week. 🥴)

▪️ Did you survive this calculation?

▪️ Have you ever taken Linear Algebra?

▪️ Leave your responses down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #46.

Cheers!

The Younge Lady

Weekly Math Problem! #14

Matrix Operations. Ahh...matrices. Just when you thought basic math operations couldn't get more complicated...enter numbers in block form. 😅🤦🏿‍♀️ In my years of doing mathematics, I see that every area of mathematics has practical applications. The truth is, most people won't learn math at the levels where such applications exist. Also for the people who do go on learn math at those levels where these applications exist, accompanying software is also learned. Doing that type of math by hand just isn't efficient. Linear Algebra has some cool applications but, before you can learn the math needed for such applications, you have to start with the basics.  

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  How to perform basic matrix operations

    

Here is WMP! #14...

Happy solving! 

Check back on Friday, August 7th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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In colored fashion, here's how the matrix calculations go:

Matrix multiplication is definitely tricky but, with a keen eye, a pattern can be seen. Yeah...don't have much to say this week; it's been a hectic one. In an upcoming WMP, I'll do a problem from the math journey I took this summer. 👀 Wondering what it is? You should stick around to find out. 😃

◾️ Have you ever done linear algebra or learned about matrix theory before?? 

◾️ If so, how long ago and what was your experience like?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Next up...WMP! #15 😏

Cheers!

The Younge Lady

Weekly Math Problem! #4

System of Linear Equations. Typically, when you've encountered this topic in learning math, you've either encountered a system of two linear equations with two unknowns (from an algebra course), or a system of three linear equations with three unknowns (from a college algebra course). I don't recall ever having to solve a system of four linear equations with four unknowns, however, when I saw it I knew that I would be able to solve it. I knew what I needed to do.

Depending on what level of math you've completed, you're aware that you have a few options when it comes to solving a system of four linear equations with four unknowns. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Methods for solving systems of linear equations

    ✔️  How to row reduce a matrix

Here goes WMP! #4... 

Happy solving! Check back on Friday, May 29th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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So, I thought it would be cool to solve this problem two ways--using the elimination method and by using Guass-Jordan elimination on the corresponding augmented matrix--just to compare how they look and see how many steps each requires to solve in completion.

Elimination Method

Guass-Jordan Elimination Method

The elimination method definitely required less steps as compared to the Guass-Jordan elimination method. As you can see, when done correctly, either method will yield the same solution. 👍🏿 The substitution method can also be used. I must say, there was something strangely sort of calming from row reducing the matrix. But then again, I'm not your typical person. 🤷🏿‍♀️ I definitely plan on having a problem from linear algebra for an upcoming WMP!  

 

◾️ Did my solution match yours?? 

◾️ Which method do you prefer when solving a system of four linear equations with four variables??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you soon for WMP! #5 😉😁

Cheers!

The Younge Lady