MATH HOMEWORK PROBLEM! SO FUN!!! VERY MUCH FUNN!!#E
WOOHOO
So,
Question,
We have 3 vectors <1, 3, 2> <0, 1, 1>
What values of x make the vectors linearly independent?
To answer that, we’ll start off with an easier question. What does being linearly independent even mean?
Well,
If you read my last blog you’ll remember that we can create new vectors by stretching and adding other vectors together. A vector is a ~linear combination~ of other vectors if it can be written in the form v1 = a*v2 + b*v3 where v1, v2, and v3 are all vectors and a and b are real numbers. A set of vectors is then linearly independent if no vector is a linear combination of the other vectors in the set.
Cool.
Next easier question. How do we solve that?
Well, very similarly to the last blog…
WITH A MATRIX!!! WOW!
So,
We set these bad boys up (vertically) in a matrix like so
And begin the process of row reduction just like we did last time [1]. First, we begin by subtracting 3 times the first row from the second.
Then we subtract 2 times the first row from the third.
Then finally we subtract the second row from the third.
The point of this seemingly arbitrary arithmetic is to make a little triangle of zeros in the bottom left corner. Why do we do this? Kinda just because it works, but mostly because we want each column to contain the first nonzero digit in a row with only zeros below it [2]. Strange, but good for us.
What’s important to understand about this process is that the vectors are linearly independent as long as the bottom right value is anything but zero [3]. Which means we have everything we need to actually answer the problem. As long as 2x+1 doesn’t equal 0 we’re linearly independent, so a little algebra tells us that x can be any real number except -0.5.
BAM!
The problem’s solved. Easy money, simple but fun, very cool.
[1] – I realize that I sorta handwaved what row reduction is or why it works last time, but to tell you the truth I only have a very fuzzy understanding of why it actually works, so it would take a VERY long time to explain, and that is not the point right now.
[2] – this is a laughably bad explanation
[3] – trust me bro [4]
[4] – I could actually do a proof of this for next week