# Change of Basis: Eigenvectors

For the problem, refer to the linear transformation T: R^3 --> R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z).

Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.

https://brainmass.com/math/linear-algebra/change-basis-eigenvectors-15922

#### Solution Preview

T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z)

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

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

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

Then the matrix is:

[2 0 2]

A= [1 -1 1]

[2 0 2]

We must find the eigenvalues of A and then find the eigenvectors of it:

The eigenvalues are 4, -1, 0. Then we have:

lambda=4 --->

...

#### Solution Summary

A change of basis matrix is found using eigenvectors. All work is shown in the solution.

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