Question

Linear algebra question

Answer 1

Apparantly, I have to find an orthogonal basis, but that is confusing me.

Answer 2

Cross product will find the orthogonal basis...
But I'm sure in this case (where we are doing linear algebra) they want you to find something like the nullspace... or was it the left nullspace?

Answer 3

I also notice the member of what? The \(a,b\in\)???

Answer 4

member of the real number I would assume.

Answer 5

@e.mccormick The \(\mathbb{R}\)s got removed.

Answer 6

Nullspace is just the solution space. That's not what we are asked.

Answer 7

I have to use projections afterwards somehow.

Answer 8

@Dido525 Remember the fundamental theorem of linear algebra? It tells you that row space and column space are same dimension, and about their orthogonality.

Answer 9

Yah, usually reals. But I know one prof that works in Q....

Answer 10

Then I would just cross w1 and w2 to find another vector in R^3 right?

Answer 11

That's one way to do it, which works okay in this example... but they probably want you to do that whole \[
A^TAx=b
\]thing or something.

Answer 12

Eww...

Answer 13

Okay so when I find that third vector, What do I do then?

Answer 14

Okay so a third vector is (0,0,1) which I got using the cross product.

Answer 15

Okay so here is what my book is saying:
When there is no solution to \[
Ax=b
\]Then you can use\[
A^TA\widehat{x}=A^Tb
\]And \(\widehat{x}\) is the answer with least error.

Answer 16

This is probably what they want you to use.

Answer 17

No... Apparantly I have to use projections somehow...

Answer 18

So finding the minimum projection value?

Answer 19

Erm...

Answer 20

Would that be least squares analysis on the prthogonal projection distance?

Answer 21

We DO NOT do least squares approximation in this course.

Answer 22

Not sure how to project a vector onto a basis other than the method I showed you.
I do know how to project a vector onto another vector.

Answer 23

OK. Hmm. No clue then.

Answer 24

Allright. I guess I will ask my professor. Thanks anyways guys!

Answer 25

Could you give him a medal too wio?

Answer 26

I am not concerened with medals.

Answer 27

Fair enough :) .

Answer 28

http://www.cliffsnotes.com/study_guide/Projection-onto-a-Subspace.topicArticleId-20807,articleId-20792.html

Answer 29

Seems that you would just add the projections onto each basis vector.

Answer 30

I find it fun and it forces me to remember things. I have even had to look up a few terms that were not in use when I learned Geometry.... like apothem. I am sure they existed, but they just did not bother with them in ages past.

Answer 31

Do you know how to project a vector onto another vector?

Answer 32

Yeah I guess you add them. Tat's what my professor said too but how many times do I add them?

Answer 33

Yeah I know.

Answer 34

There are two basis vectors... so you do two projections which are added.