Linear Algebra (Symmetric Matrix)

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Discussion Overview

The discussion revolves around the properties of a 3x3 symmetric matrix, specifically focusing on the null space, column space, row space, and left null space. Participants explore how to determine the bases and dimensions of these spaces given the dimension of the null space.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant, MoBaT, seeks guidance on how to find the bases of the column space, row space, and left null space with only the dimension of one vector known.
  • Another participant suggests that any linearly independent basis can be chosen, implying flexibility in the selection of bases.
  • A different participant claims to know the dimensions of the column space and row space, stating they are both 2, while the dimension of the left null space is 1, and provides a basis for the row space.
  • MoBaT expresses confusion about how the basis for Row(A) was determined.
  • A later reply indicates that the participant figured out how to derive the bases by using the identity matrix to fill in the remaining information.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints regarding the determination of bases and dimensions, with some participants providing specific answers while others express uncertainty or seek clarification. No consensus is reached on the methodology for finding the bases.

Contextual Notes

Participants rely on the properties of symmetric matrices and the relationships between the various spaces, but the discussion does not clarify the assumptions or specific steps taken to arrive at the proposed bases.

MoBaT
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A 3x3 symmetric matrix has a null space of dimension one containing the vector (1,1,1). Find the bases and dimensions of the column space, row space, and left null space.

I understand how to get the Dim of Col(A), Row(A), and Nul(A^T) but how do i get the bases with just knowing the dimension of one vector? How should I approach this?
 
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Hey MoBaT.

Can you pick any linearly independent basis?
 
chiro said:
Hey MoBaT.

Can you pick any linearly independent basis?

doesn't say anything against it. I know what the answer is because he gave it to us. It was like:

Dim of Col(A) = 2, dim Row(A) = 2, Dim Nul(A^T) = 1
Basis Row(A) = {[-1 1 0], [-1 - 1]}. Because A is symmetric, Col(A) = Row(A) and Nul(A^T) = Nul(A).

I understand everything after the Basis row(A) but do not understand how he got that Row(A)
 
Found out how to do it. Pretty much fill in the rest of the information with the identity matrix.
 

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