[goMac]

So I have a problem in the textbook that is notated funny, I was wondering if someone in here knew how to solve it...

Find the orthogonal complement of W and give a basis for the orthogonal compliment:

W = {[x y] : 2x -y = 0}

(x and y are in the same column, but that's kind of hard to type...)

The book doesn't demonstrate how to do the problem when phrased in this form. I thought maybe I could turn it into the vector [2 1] but I don't feel like that's taking me down the right path.

Anyone?

[ibook_steve]

Over 13 years since I did anything like that...wish I could help. I kind of liked doing that stuff back at school.

Steve

[Ghoser777]

Orthogonal means perpendicular. So my assumption is that's just asking for the set of lines perpendicular to 2x - y = 0. That set would happen to be x + 2y = a, where a is any real number.

[Ghoser777]

Here's a wikipedia article related to it (maybe a little too technical):

http://en.wikipedia.org/wiki/Orthogonal_complement

[goMac]

Originally Posted by Ghoser777 

Orthogonal means perpendicular. So my assumption is that's just asking for the set of lines perpendicular to 2x - y = 0. That set would happen to be x + 2y = a, where a is any real number.

The book gives x + 2y = 0 as the answer, so you're right.

This is going to be a really dumb question, as I'm pretty bad at picking this stuff up, but what methodology did you go through to get the orthogonal function?

In addition, the book gives the answer for the basis as [-2 1], which I can't figure out how it came up with.

Sorry, I just don't pick up this stuff well...

[Ghoser777]

Well, I cheated. I haven't taken linear algebra in a long time, but I know how to find a line perpendicular to another line. In the two dimensional case, you just need the slope's to be opposite recipricols. If you solve your first equation for y, you'll get y = 2x. (slope is 2). If you solve the answer for y, you'll get y = (-1/2x). (slope of -1/2).

Now, the correct way to do it is probably to use the dot product (I have no idea how far you are in your class). The dot product of two vectors is defined as the sum of the products of the corresponding parts of two vectors. i.e.

[a b] dot [c d] = ac + bd.

You have the vector [2 -1] as your given equation. If you dot that vector with another vector [a b], the sum has to be 0 for the vectors to be perpendicular. There are infinitely many vectors that will satisfy that property, but the simplest one is [-1 2].

I haven't taken linear algebra in a long time, so there probably is another way to do this.

[Ghoser777]

Oh yeah, and the dot product is the same thing as the inner product (of that was confusing you - it's a different name for the same thing).