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Monthly Archives: January 2014
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
Posted in linear algebra
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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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