Worksheet: Gram–Schmidt Process

In this worksheet, we will practice finding the orthonormal basis of a vector space using the Gram–Schmidt process.

Q1:

Find an orthonormal basis of eigenvectors for the matrix

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• B
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• E

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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Q4:

Find an orthonormal basis of eigenvectors for the matrix

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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: 2, 6, and 12, eigenvectors:
• Deigenvalues: 6, 12, and 18, eigenvectors:
• Eeigenvalues: 6, 12, and 18, eigenvectors:

Q6:

Find an orthonormal basis of eigenvectors for the matrix

• A
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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 Gram–Schmidt process to the vectors , , and to find an orthonormal basis for their span.

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Q9:

Apply the Gram–Schmidt process to the vectors , , and to find an orthonormal basis for their span.

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Q10:

Apply the Gram–Schmidt process to the vectors , , and to find an orthonormal basis for their span.

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Q11:

Which of the following is an orthonormal set of vectors in ?

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Q12:

The set is a subspace of . Find an orthonormal basis for this subspace.

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Q13:

Fill in the blank. Columns of an matrix are an orthonormal basis for , if and only if is a matrix.

• Anormal
• Bunitary
• Csymmetric
• Dsquare

Q14:

Fill in the blank. The of orthogonal vectors is -dimensional.

• Acollection
• Bkernel
• Ctransformation
• Dspan