# Worksheet: Orthonormal Basis

In this worksheet, we will practice finding the eigenvalues and the orthonormal basis of eigenvectors for a matrix.

Q1:

Find an orthonormal basis of eigenvectors for the matrix

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Q2:

Find an orthonormal basis of eigenvectors for the matrix

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• E

Q3:

Find an orthonormal basis of eigenvectors for the matrix

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• E

Q4:

Find an orthonormal basis of eigenvectors for the matrix

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• E

Q5:

Given that find the eigenvalues and an orthonormal basis of eigenvectors for .

• Aeigenvalues: 2, 6, and 12, eigenvectors:
• Beigenvalues: 6, 12, and 18, eigenvectors:
• Ceigenvalues: 6, 12, and 18, eigenvectors:
• Deigenvalues: 6, 12, and 18, eigenvectors:
• Eeigenvalues: 2, 6, and 12, eigenvectors:

Q6:

Find an orthonormal basis of eigenvectors for the matrix

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Q7:

The two level surfaces and intersect in a subspace of . Find a basis for this subspace, and then find an orthonormal basis for this subspace.

• Abasis: , , orthonormal basis: ,
• Bbasis: , , orthonormal basis: ,
• Cbasis: , , orthonormal basis: ,
• Dbasis: , , orthonormal basis: ,
• Ebasis: , , orthonormal basis: ,

Q8:

Apply the Gramâ€“Schmidt process to the vectors , , and to find an orthonormal basis for their span.

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