(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

3. The attempt at a solution

T(1,0,0) = (3,-1,0)

T(0,1,0) = (0,1,0)

T(0,0,1) = (-1,2,4)

Thus, we have the matrix,

[itex]\left| \begin{array}{ccc}

3 &0&-1 \\

-1&1&2 \\

0&0&4 \end{array} \right|[/itex]

[itex]Δ_T (t) = det( \left| \begin{array}{ccc}

3 &0&-1 \\

-1&1&2 \\

0&0&4 \end{array} \right| - tI)[/itex]

I have this equaling: -(t-4)(t-3)(t-1), which is the characteristic polynomial. The roots are the eigenvalues, which are 4,3,1.

To compute the eigenvectors:

When t=4, we have,

-x-z=0

-x-3y+2z=0

0z=0

Which implies that eigenvectors are multiples of (-1,1,1).

When t=3, we have,

-z=0

-x-2y=0

z=0

Which implies that eigenvectors are multiples of (-1,2,0)

When t=1, we have,

2x-z=0

-x+2z=0

3z=0

Which implies that eigenvectors are multiples of (0,1,0).

T is diagonizable because (-1,1,1),(-1,2,0),(0,1,0) are lin. indep.

First of all, are these correct? Also, how does one determine eigenspaces? That's the part I feel that I really don't know.

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# Homework Help: Eigenvalues, eigenvectors, and eigenspaces

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