Very Basic Simple Breakdown of Linear Algebra

Series: Linear Algebra

If you are currently learning linear algebra for any reason be it for school or just for the love of learning this is for you! If you feel overwhelmed with the sheer amount of concepts in Linear Algebra this series is for you. I will try to explain the mysterious world of Linear Algebra in a way that is digestible for even the casual reader with a limited amount of mathematical background.

The first post in a series about Linear Algebra I will be talking about a topic of Gaussian elimination / Row Echelon Form.

• Matrix

Question: What the heck is a Matrix anyway?! Are you talking about the movie?!!

Answer: No silly 😊 It is simply a rectangle array of numbers! It has a number of rows and columns that store numbers in them! That’s it!

• Row Echelon Form

Question: How can I tell if my matrix is in row echelon form?

Answer: Two specifications must be true:

1. All nonzero (any number except 0) rows with at least one nonzero element (number) has to be above a row that has ALL ZEROS
2. The leading coefficient or called pivot of a nonzero row HAS to be right of the leading coefficient of the row above it

Figure 1. Matrix in both Row Echelon Form and Reduced Row Echelon Form

If you look at Figure 1 in the last row:

All of the elements in this row are ALL 0! So the 1st specification is TRUE!

Nice!!!

If you look at Figure 1 in the first row:

Figure 2. Row 1 in Figure 1

Their is a 1 in the first column… cool!

Figure 3. Row 2 in Figure 1 where in the second column their is a number 1

Their is a 1 in the second column, which is right of the leading coefficient of the row above it.

If you look at the third row:

Figure 4. Row 3 in Figure 1 where in the third column their is a number 1

Their is a 1 in the third column cool!

Figure 5. Row 4 in Figure 1 where in the forth column their is a number 1

But their is a 1 in the forth column, which is right of the leading coefficient of the row above it.

• Reduced Row Echelon Form

Question: How can I tell if my matrix is in reduced row echelon form?

Answer: The following HAS to be true:

1. It is in row echelon form

• Look at figure 1.

1. The leading coefficient in each nonzero row has to be the number 1

• Look at figure 1.

1. Each column that has a leading 1 has to have a zero everywhere else

• Look at figure 1.

I know this is brief it is suppose to be 😍 to familiar ourselves with the vocabulary in linear algebra. The next post in the Linear Algebra series will be about the difference between dot product and cross product since this is used a ton in Linear Algebra.

Resources:

This first website is an ultimate life saver if you need a reference to figure out a problem that you are currently facing or if you are trying to ace your Linear Algebra exam and is able to use the internet:

http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

This next resource is a YouTube video from Lorenzo Sadun who brakes brakes down both Row Echelon Form and Reduced Row Echelon Form pretty easily:

https://www.youtube.com/watch?v=l69YjkuUym0

Wikipedia:

https://en.wikipedia.org/wiki/Row_echelon_form

Khan Academy:

https://www.khanacademy.org/