/sci/ - Science & Math

Also, whoever helps me can ask anything of me (within reason) and I will do it. OP WILL DELIVER.

I just need to pass this fucking class.

>>4616266
Think what it means for a vector to be in a plane determined by a basis. If u is in H then it must be expressed as a linear combination of the basis vectors of H.

nigga what the fuck is a hyperplane

>>4616309

Right, I kind of understand the jist of it, I'm just not sure how to show it using row ops.

Let's just pretend this shit is the putnam problem of the day.

If I wrote the equations as:

1: 2W+2X+5Y+2Z= 0

etc

I could show u isn't in the plane right?

It's just that that isn't suing row ops

>>4616266
as for c) , I'm not sure exactly what form you want the hyperplane to be in, but it might be something like:
N dot (x - b1) = 0 , where N is a vector normal to the plane.

To find N you just need to use the cross product on b1 and b2, then since they span H it follows that every vector in H will be orthogonal to N, and you are asked to verify this for u and w (ie: N dot u is not 0, but N dot w is 0)

Procedure:
1. To show the fourth vector is in not in the hyperplane, take the four vectors and make a matrix. set this matrix equal to the 0 vector in R4, which means the vector [0 0 0 0 ]^T. So now you have Ax=b, A is the matrix out of the four vectors, b is the 0 vector. If the fourth vector is not in H, the only x that you will get that will solve the matrix is when x=0.

2. The same as before, but this time x can be some vector that is not 0 in order to solve Ax=b.

3. I think this part is basically to find the orthogonal set? Use Gram-schmidt and find the vectors. if you don't know what that is, just ask.

>>4616324
Recall what Ax = b means .
It means that A acts on x to produce the vector b, OR
that b can be written as a linear combination of the columns of A.

>>4616266
For (a) row reduce the matrix formed by v_1, v_2, and v_3 and show that your vector u is not formed by a linear combination of the row reduced form of P; that is, show that u =/= av_1 + bv_2 + cv_3 for a,b,c in your field (likely the real numbers).
(b) follows by showing that w = av_1+bv_2+cv_3
As for part (c) find the normal vectors to each basis vector and combine then find the vector orthogonal to the plane. That will give you your normal. Then just dot it. Simple.

>>4616372
>>4616350
>>4616371
>>4616364

I love you guys so much, I'm working the problem now I'll post if I run into a speedbump, however for part C, I don't think it uses Gram-Schmidt as that was explicitly covered on a separate question on the exam.

>>4616350
oups btw i noticed that in my picture i wrote N = b1 x b2, but it should be N = b2 x b1

>>4616381
i don't exactly know what the dot product form is. but if the goal is to find the orthogonal set, you can manually compute the orthogonal vectors by projection, but the gram schmidt is exactly that, and i don't know how else you could get an orthogonal basis

>>4616371

Alright, I reduced it down to RREF, with a row of 0's on the bottom, How should this play into u and w?

Sorry, I'm fucking retarded guys.

>>4616403
if you used all four vectors, and you got a row of 0's, then you basically have a linearly dependent set. that means the vector is already in the "cube" spanned by the first three vectors.

>>4616410

I only used the original 3, should I augment it with u? or put u right into the matrix?

Okay, I augmented the v matrix with u, did row ops, and the augmented side came out to this.

>>4616512
the matrix should have four vectors v1 v2 v3 and u. and you should augment the [0 0 0 0] vector. so its going to look like [v1 v2 v3 u | 0]