Geometric description of the nullspace

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Homework Help Overview

The discussion revolves around finding a basis for the nullspace of a given matrix A and providing a geometric description of that nullspace. The matrix A is presented in a specific form, and the original poster expresses uncertainty about how to describe the geometric nature of the nullspace based on the basis they have identified.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the basis vectors identified for the nullspace and question how to interpret these vectors geometrically. There is an exploration of what it means for vectors to span different types of spaces, such as lines or planes.

Discussion Status

The discussion is ongoing, with participants seeking clarification on the geometric interpretation of the nullspace. Some guidance has been offered regarding the nature of linearly independent vectors and their relationship to dimensionality, but a consensus on the geometric description has not yet been reached.

Contextual Notes

Participants are navigating the definitions and properties of vector spaces, particularly focusing on the implications of linear independence and dimensionality in the context of the nullspace.

Tomblue
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Homework Statement


general form of solutions to Ax=b
Consider matrix A=
{[ 2 -10 6 ]
[ 4 -20 12 ]
[ 1 -5 3 ]}
Find a basis for the nullspace of A. Give a geometric description of the nullspace of A.


The Attempt at a Solution


I found the basis for the nullspace of A to be
{[-3 5]
[0 1 ]
[1 0 ]}
The thing i don't understand is how to give a geometric description of the nullspace of A. If someone could help to explain how i would start to go about doing this that would be awesome because I'm not quite sure i understand the question.
 
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Are you saying that your null space is spanned by {-3, 0, 1} and {5, 1, 0}? Your notation is a bit confusion to me; but I believe that they are asking you to describe what kind of "Space" this is. What do two vectors span?
 
I was trying to say that the vectors {-3,0,1} and {5,1,0} form the basis for the nullspace of A and that I'm not seeing how to give a geometric description of the nullspace of A.
 
How many linearly independent vectors does it take to span a line? A plane? A three-dimensional space?
 
one linearly independent vector to span a line, two linearly independent vectors to span a plane, and 3 linearly independent vectors to span a 3-dimensional space, and so forth any n vectors that span an n-dimensional space are going to be linearly independent. So i see I'm going to have 2 linearly independent vectors and therefore the dim(W) is going to be 2 dimensional like i thought. Question: Can the vectors ever be identical say that v1=v2
 
Linearly independent vectors can't be identical or even multiples of one another.
 

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