Finding a Matrix whose kernel is spanned by 2 vectors

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Homework Help Overview

The problem involves finding a matrix whose kernel is spanned by the vectors u=(1,3,2) and v=(-2,0,4). Participants are exploring various approaches to construct such a matrix.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • One participant attempted to set the vectors as a matrix and apply row reduction but expressed uncertainty about the process. Another suggested finding a vector orthogonal to both u and v to create a matrix with that vector as rows. A third participant provided a matrix formed from u, v, and an additional vector but questioned the validity of their result when checking the kernel condition.

Discussion Status

The discussion is ongoing, with participants exploring different methods to find a suitable matrix. Some guidance has been offered regarding the use of orthogonal vectors, and there is a critical examination of the validity of proposed matrices based on kernel properties.

Contextual Notes

Participants are navigating the constraints of the problem, including the requirement for the kernel to be spanned by the specified vectors and the implications of matrix operations on kernel membership.

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Homework Statement


Find a matrix whose kernel is spanned by the two vectors u=(1,3,2) and v=(-2,0,4).


Homework Equations





The Attempt at a Solution


Tried setting vectors as a matrix and rref'ing it, but didn't know where I was getting at, also tried using an augmented identity matrix with both vectors then realized I didn't know what I was doing.
 
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How about finding a vector w that's orthogonal to both u and v and making a matrix where all of the rows are w?
 
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Likes   Reactions: Sushil Khatri
k I think I got it...
Given:
u=[1,3,2]
v=[-2,0,4]

I put in the vector M(1)=[1,1,1]
M:=<u|v|M(1)>
and I get...
[1,-2,1
3,0,1
2,4,1]

I apply rref, and I get an identity matrix. Meaning that the above matrix is my answer, right?
 
That doesn't work. M*u is (-3,5,14) isn't it? If u is in the kernel M*u is supposed to (0,0,0). I don't think you heard me. Find a vector w so that u.w=0 and v.w=0. Wouldn't it work if you make a matrix with all of the rows w?
 

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