6.1-6.4 Key Exercises

Math 2331 – Linear Algebra6.1-6.4 Orthogonality

Key Exercises

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.edumath.uh.edu/∼jiwenhe/math2331

Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

6.1 Inner Product, Length & Orthogonality

Key Exercises 19–20, 27–31

The general material on orthogonal complements is essentialfor later work.

Key Exercises: 19–20, 27–31. Exercies 27–31 concern factsthat are needed later

Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

6.2 Orthogonal Sets

Key Exercises 13–14, 23–24, 26–30

The nonsquare matrices in Theorems 6 and 7 are needed forthe QR factorization in Section 6.4.

The term orthogonal matrix applies only to certain squarematrices.

Key Exercises: 13–14, 23–24, 26–30. Exercies 13–14 preparefor Section 6.3.

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

6.3 Orthogonal Projections

Key Exercises 19–24

Theorem 9 is needed for the Gram-Schmidt process.

Key Exercises: 19–24. Exercies 19–20 lead naturally into theGram-Schmidt process.

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 23 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 24 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

6.4 The Gram-Schmidt Process

Key Exercises 17–21,23

The QR factorization encapsulates the essential outcome ofthe Gram-Schmidt process, just as the LU factorizationdescribes the result of a row reduction process.

For practical use of linear algebra, the factorizations are moreimportant than the algorithms that produce them.

Key Exercises: 17–21, 23.

Jiwen He, University of Houston Math 2331, Linear Algebra 27 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 28 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

All vectors are in Rn. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 29 / 33

6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

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6.1-6.4 Key Exercises 6.1: 19–31 6.2: 13–30 6.3: 19–24 6.4: 17–23

Jiwen He, University of Houston Math 2331, Linear Algebra 33 / 33