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Worksheet: Orthonormal Basis

Q1:

Apply the Gram–Schmidt process to the vectors ( 1 , 2 , 1 ) , ( 2 , 1 , 3 ) , and ( 1 , 0 , 0 ) to find an orthonormal basis for their span.

  • A 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • B 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • C 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • D 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • E 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3

Q2:

Find an orthonormal basis of eigenvectors for the matrix

  • A 1 0 0 , 1 5 0 2 1 , 1 5 0 1 2
  • B 1 0 0 , 0 1 1 , 0 1 1
  • C 1 0 0 , 0 2 1 , 0 1 2
  • D 1 0 0 , 1 2 0 1 1 , 1 2 0 1 1
  • E 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1

Q3:

Find an orthonormal basis of eigenvectors for the matrix

  • A 1 0 5 , 1 0 3 2 , 5 2 4 1
  • B 1 0 5 , 5 6 2 1 , 5 2 6 1
  • C 1 2 6 1 0 5 , 1 1 1 3 1 0 3 2 , 1 6 0 2 5 2 4 1
  • D 1 6 1 0 5 , 3 0 1 5 5 6 2 1 , 1 3 0 5 2 6 1
  • E 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1

Q4:

Find an orthonormal basis of eigenvectors for the matrix

  • A 1 3 1 1 1 , 1 6 1 1 2 , 1 2 1 1 0
  • B 1 1 1 , 1 1 2 , 1 1 0
  • C 1 1 1 , 1 0 1 , 1 2 1
  • D 1 3 1 1 1 , 1 6 1 1 2 , 1 2 1 1 0
  • E 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1

Q5:

Find an orthonormal basis of eigenvectors for the matrix

  • A 1 0 0 , 1 5 0 2 1 , 1 5 0 1 2
  • B 1 0 0 , 0 1 1 , 0 1 1
  • C 1 0 0 , 0 2 1 , 0 1 2
  • D 1 0 0 , 1 2 0 1 1 , 1 2 0 1 1
  • E 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1

Q6:

Apply the Gram–Schmidt process to the vectors ( 3 , 4 , 0 ) , ( 7 , 1 , 0 ) , and ( 1 , 7 , 1 ) to find an orthonormal basis for their span.

  • A 3 5 4 5 0 , 3 5 4 5 0 , 0 0 1
  • B 4 5 3 5 0 , 4 5 3 5 0 , 0 0 1
  • C 4 5 3 5 0 , 3 5 4 5 0 , 0 0 1
  • D 3 5 4 5 0 , 4 5 3 5 0 , 0 0 1
  • E 3 5 4 5 0 , 3 5 4 5 0 , 0 0 1

Q7:

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

  • Aeigenvalues: 2, 6, and 12, eigenvectors: 1 6 1 1 2 2 , 1 2 1 1 0 1 2 , 1 3 1 1 1 6
  • Beigenvalues: 6, 12, and 18, eigenvectors: 1 6 1 1 2 1 2 , 1 2 1 1 0 6 , 1 3 1 1 1 1 8
  • Ceigenvalues: 6, 12, and 18, eigenvectors: 1 6 1 1 2 6 , 1 2 1 1 0 1 2 , 1 3 1 1 1 1 8
  • Deigenvalues: 6, 12, and 18, eigenvectors: 1 6 1 1 2 6 , 1 2 1 1 0 1 2 , 1 3 1 1 1 1 8
  • Eeigenvalues: 2, 6, and 12, eigenvectors: 1 6 1 1 2 2 , 1 2 1 1 0 6 , 1 3 1 1 1 1 2

Q8:

Find an orthonormal basis of eigenvectors for the matrix

  • A 1 3 1 1 1 , 1 2 1 1 0 , 1 6 1 1 2
  • B 1 1 1 , 1 1 0 , 1 1 2
  • C 1 1 1 , 1 0 1 , 1 2 1
  • D 1 3 1 1 1 , 1 2 1 1 0 , 1 6 1 1 2
  • E 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1

Q9:

The two level surfaces 2 𝑥 + 3 𝑦 𝑧 + 𝑤 = 0 and 3 𝑥 𝑦 + 𝑧 + 2 𝑤 = 0 intersect in a subspace of 4 . Find a basis for this subspace, and then find an orthonormal basis for this subspace.

  • Abasis: 2 5 1 1 0 , 7 1 0 1 1 , orthonormal basis: 1 1 5 6 1 6 6 1 1 3 0 6 0 , 4 6 3 1 3 5 6 2 0 9 1 1 2 5 4 6 2 0 9 1 3 3 0 6 2 0 9 5 2 0 9 6 2 0 9
  • Bbasis: 2 5 1 1 0 , 7 1 0 1 1 , orthonormal basis: 1 1 5 6 1 6 6 1 1 3 0 6 0 , 4 6 3 1 3 5 6 2 0 9 1 1 2 5 4 6 2 0 9 1 3 3 0 6 2 0 9 5 2 0 9 6 2 0 9
  • Cbasis: 2 5 1 1 0 , 7 1 0 1 1 , orthonormal basis: 1 1 5 6 1 6 6 1 1 3 0 6 0 , 4 6 3 1 3 5 6 2 0 9 1 1 2 5 4 6 2 0 9 1 3 3 0 6 2 0 9 5 2 0 9 6 2 0 9
  • Dbasis: 2 5 1 1 0 , 7 1 0 1 1 , orthonormal basis: 1 1 5 6 1 6 6 1 1 3 0 6 0 , 4 6 3 1 3 5 6 2 0 9 1 1 2 5 4 6 2 0 9 1 3 3 0 6 2 0 9 5 2 0 9 6 2 0 9
  • Ebasis: 2 5 1 1 0 , 7 1 0 1 1 , orthonormal basis: 1 6 6 1 1 5 6 1 1 3 0 6 0 , 4 6 3 1 3 5 6 2 0 9 1 1 2 5 4 6 2 0 9 1 3 3 0 6 2 0 9 5 2 0 9 6 2 0 9

Q10:

Apply the Gram–Schmidt process to the vectors ( 1 , 2 , 1 , 0 ) , ( 2 , 1 , 3 , 1 ) , and ( 1 , 0 , 0 , 1 ) to find an orthonormal basis for their span.

  • A 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • B 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • C 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • D 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • E 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3