Math 2331 { Linear Algebrajiwenhe/math2331/lectures/sec6_1-4_ex.p… · 6.1-6.4 Key Exercises 6.1:...
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6.1-6.4 Key Exercises Math 2331 – Linear Algebra 6.1-6.4 Orthogonality Key Exercises Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 33
Transcript of Math 2331 { Linear Algebrajiwenhe/math2331/lectures/sec6_1-4_ex.p… · 6.1-6.4 Key Exercises 6.1:...
6.1-6.4 Key Exercises
Math 2331 – Linear Algebra6.1-6.4 Orthogonality
Key Exercises
Jiwen He
Department of Mathematics, University of Houston
[email protected]/∼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.
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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
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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
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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 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.
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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
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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.
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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.
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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
6.4 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.
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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 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.
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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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