Suppose a is the matrix for t: r3 → r3 relative to the standard basis. find the diagonal matrix a' for t relative to the basis b'.

Step-by-step explanation:

The first that you must ask is whether or not the matrix is a diagonalizable matrix. If so, you have to find a basis of consisting of eigenvectors of the matrix . To do that first find the eigenvalues of the matrix and the the corresponding eigenspaces to that eigenvalues. This will give you the basis of consisting of eigenvectors of A. Next apply the change of coordinates using the matrix of change of coordinates whose columns are the vectors on the new basis. So we have that

is the matrix of T respect to new basis consisting of eigenvectors of T, and it is a diagonal matrix, and its elements on the diagonal are the eigenvalues.

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step-by-step explanation: