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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.1.2
Exercise 3.1.2. For give an example of linearly independent vectors that are not mutually orthogonal, as well as mutually orthogonal vectors that are not linearly independent. Answer: The vectors and are linearly independent, since the second vector cannot be expressed … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.1
Exercise 3.1.1. For and what is the length of each vector and their inner product? Answer: We have and so that and . The inner product of and is then Note that and are thus orthogonal. NOTE: This continues a … Continue reading
Completing Chapter 2 of Linear Algebra and Its Applications
Yesterday I posted the final worked-out solution for the exercises from chapter 2 of Gilbert Strang’s Linear Algebra and Its Applications, Third Edition. My first post for chapter 2 was for exercise 2.1.1 almost exactly 29 months ago. This is … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.33
Review exercise 2.33. Consider the following factorization: a) What is the rank of ? b) Find a basis for the row space of . c) Are rows 1, 2, and 3 of linearly independent: true or false? d) Find a … Continue reading
Posted in linear algebra
Tagged basis, column space, dimension, factorization, general solution, linear independence, nullspace, rank, row space
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Linear Algebra and Its Applications, Review Exercise 2.32
Review exercise 2.32. a) Find the subspace of such that for any vector in the subspace we have . b) Find a matrix for which this subspace is the nullspace. c) Find a matrix for which this subspace is the … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.31
Review exercise 2.31. Consider the rank-one matrix . Under what conditions would ? Answer: In order for to exist must be a square matrix; otherwise we could not multiply by since the number of columns of the first matrix would … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.30
Review exercise 2.30. Suppose that the matrix is a square matrix. a) Show that the nullspace of contains the nullspace of . b) Show that the column space of contains the column space of . Answer: a) Suppose is in … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.29
Review exercise 2.29. The following matrices represent linear transformations in the – plane with and as a basis. Describe the effect of each transformation. Answer: When the matrix is applied to the vector we obtain When the matrix is applied … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.28
Review exercise 2.28. a) If is an by matrix with linearly independent rows, what is the rank of ? The column space of ? The left null space of ? b) If is an 8 by 10 matrix and the … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.27
Review exercise 2.27. Find bases for each of the following matrices: Answer: If we put in echelon form (by exchanging rows 3 and 4) the resulting matrix would have pivots in columns 1, 2, and 4. Columns 1, 2, and … Continue reading
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