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Monthly Archives: December 2013
Linear Algebra and Its Applications, Review Exercise 2.26
Review exercise 2.26. State whether the following statements are true or false: a) For every subspace of there exists a matrix for which the nullspace of is . b) For any matrix , if both and its transpose have the … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.25
Review exercise 2.25. Suppose that is a linear transformation from to itself, and transforms the point to the point . What does the inverse transformation do to the point ? Answer: The effect of is to reverse the effect of … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.24
Review exercise 2.24. Suppose that is a 3 by 5 matrix with the elementary vectors , , and in its column space. Does has a left inverse? A right inverse? Answer: Since , , and are in the column space … Continue reading
Posted in linear algebra
Tagged column space, elementary vectors, left inverse, right inverse
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Linear Algebra and Its Applications, Review Exercise 2.23
Review exercise 2.23. Given any three vectors , , and in find a matrix that transforms the three elementary vectors , , and respectively into those three vectors. Answer: When multiplying by only the entries in the first column of … Continue reading
Posted in linear algebra
Tagged elementary vectors, invertible, linear transformation
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Linear Algebra and Its Applications, Review Exercise 2.22
Review exercise 2.22. a) Given what conditions must satisfy in order for to have a solution? b) Find a basis for the nullspace of . c) Find the general solution for for those cases when a solution exists. d) Find … Continue reading
Posted in linear algebra
Tagged basis of column space, basis of nullspace, conditions on b, general solution, rank
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Linear Algebra and Its Applications, Review Exercise 2.21
Review exercise 2.21. Consider an by matrix with the value 1 for every entry. What is the rank of such a matrix? Consider another by matrix equivalent to a checkerboard, with if is even and if is odd. What is … Continue reading
Linear Algebra and Its Applications, Review Exercise 2.20
Review exercise 2.20. Consider the set of all 5 by 5 permutation matrices. How many such matrices are there? Are the matrices linearly independent? Do the matrices span the set of all 5 by 5 matrices? Answer: An example member … Continue reading
Posted in linear algebra
Tagged linear independence, permutation matrices, spanning set, subspace
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Linear Algebra and Its Applications, Review Exercise 2.19
Review exercise 2.19. Consider the set of elementary 3 by 3 matrices with ones on the diagonal and at most one nonzero entry below the diagonal. What subspace is spanned by these matrices? Answer: An example member of this set … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.18
Review exercise 2.18. Suppose that is an by matrix with rank . Show that if then . Answer: Since the rank of is we know that the columns of are linearly independent and that the inverse exists. We then have … Continue reading
Posted in linear algebra
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