MHB Basis for the eigenspace corresponding

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To find the basis for the eigenspace corresponding to the eigenvalue λ = 3, one must solve the equation (A - λI)v = 0. The specific matrix to analyze is given as [1 2 3; -1 -2 -3; 2 4 6]. By inspecting the rank of this matrix, the rank-nullity theorem can be applied to determine the number of basis vectors for the null space. This method provides a systematic approach to identifying the eigenspace and its basis. Understanding these concepts is crucial for solving problems related to eigenvalues and eigenspaces effectively.
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saqifriends said:
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Hi saqifriends, :)

I have outlined the method to do this kind of problems http://www.mathhelpboards.com/threads/1270-basis-for-each-eigenspace?p=6086&viewfull=1#post6086 Since you have been given a particular eigenvalue, find the eigenspace corresponding to that eigenvalue. Then find a basis for that eigenspace.

Kind Regards,
Sudharaka.
 
as a slight nudge towards the answer, solve the system:

(A - λI)v = 0. in this case, λ = 3, so you must find the null space of the matrix:

$\begin{bmatrix}1&2&3\\-1&-2&-3\\2&4&6 \end{bmatrix}$

the rank of this matrix should be obvious upon inspection, and the rank-nullity theorem then tells you how many basis vectors you should have for the null space.
 
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