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 3000321201232.

  • A100,12011,12011
  • B100,021,012
  • C13111,12101,16121
  • D100,15021,15012
  • E100,011,011

Q2:

Find an orthonormal basis of eigenvectors for the matrix 200051015.

  • A13111,12101,16121
  • B100,011,011
  • C100,15021,15012
  • D100,021,012
  • E100,12011,12011

Q3:

Find an orthonormal basis of eigenvectors for the matrix 177471744414.

  • A111,101,121
  • B13111,12101,16121
  • C13111,16112,12110
  • D13111,16112,12110
  • E111,112,110

Q4:

Find an orthonormal basis of eigenvectors for the matrix 111411144414.

  • A13111,12110,16112
  • B111,101,121
  • C111,110,112
  • D13111,12101,16121
  • E13111,12110,16112

Q5:

Given that 𝐴=131411344410, find the eigenvalues and an orthonormal basis of eigenvectors for 𝐴.

  • Aeigenvalues: 2, 6, and 12, eigenvectors: 161122,121106,1311112
  • Beigenvalues: 6, 12, and 18, eigenvectors: 161126,1211012,1311118
  • Ceigenvalues: 2, 6, and 12, eigenvectors: 161122,1211012,131116
  • Deigenvalues: 6, 12, and 18, eigenvectors: 1611212,121106,1311118
  • Eeigenvalues: 6, 12, and 18, eigenvectors: 161126,1211012,1311118

Q6:

Find an orthonormal basis of eigenvectors for the matrix 533015851530151456158515615715.

  • A105,1032,5241
  • B13111,12101,16121
  • C16105,30155621,1305261
  • D126105,11131032,16025241
  • E105,5621,5261

Q7:

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

  • ABasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • BBasis: 25110, 71011, orthonormal basis: 1661156113060, 463,135620911,25462091330620952096209
  • CBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • DBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • EBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209

Q8:

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.

  • A666366,321022522,731531533
  • B666366,321022522,731531533
  • C666366,321022522,731531533
  • D666366,321022522,731531533
  • E666366,321022522,731531533

Q9:

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.

  • A45350,35450,001
  • B45350,45350,001
  • C35450,35450,001
  • D35450,35450,001
  • E35450,45350,001

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.

  • A6663660,66269561869,51111111113331711133322111333
  • B6663660,66269561869,51111111113331711133322111333
  • C6663660,66269561869,51111111113331711133322111333
  • D6663660,66269561869,51111111113331711133322111333
  • E6663660,66269561869,51111111113331711133322111333

Q11:

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

  • A200,010,003
  • B131313,12012
  • C111,101
  • D010,001,000

Q12:

The set 𝑉={(𝑥,𝑦,𝑧)2𝑥+3𝑦𝑧=0} is a subspace of . Find an orthonormal basis for this subspace.

  • A255055,37035701437070
  • B255055,37035701437070
  • C255066,703570147070
  • D550255,37035701437070
  • E550255,37035701437070

Q13:

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

  • Anormal
  • Bunitary
  • Csymmetric
  • Dsquare

Q14:

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

  • Acollection
  • Bkernel
  • Ctransformation
  • Dspan

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