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Monthly Archives: November 2013
Linear Algebra and Its Applications, Review Exercise 2.17
Review exercise 2.17. Suppose that is a vector in and that for all . Show that must be the zero vector. Answer: For and we have SinceĀ for all we must have for each of the elementary vectors , , … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.16
Review exercise 2.16. For each of the cases in review exercise 2.15, describe the relationship among the rank of , the number of rows , and the number of columns . Answer: (i) For the matrix we have , , … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.15
Review exercise 2.15. For each of the following, find a matrix for which i) there is either no solution or one solution to depending on ii) there are an infinite number of solutions to for all iii) there is either … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.14
Review exercise 2.14. Do the three vectors , , and form a basis for the vector space ? Answer: In order for the three vectors to be a basis for they must be linearly independent. We can test this by … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.13
Review exercise 2.13. For the matrix find its triangular factors and describe the conditions under which the columns of are linearly independent. Answer: We start elimination by subtracting 1 times the first row from the second row, with We next … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.12
Review exercise 2.12. The matrix is by and has rank . What is the dimension of its nullspace? Answer: The dimension of the nullspace is the number of columns of minus the rank of , or . NOTE: This continues … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.11
Review exercise 2.11. a) Given the following matrix find its rank and a basis for the nullspace. b) Are the first three rows of a basis for the row space of ? Are the first, third, and sixth columns of … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.10
Review exercise 2.10. Given the set of all linear transformations from to define operations for scalar multiplication and vector addition that will make the set a vector space. What is the dimension of the resulting vector space? Answer: Let and … Continue reading
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Linear Algebra and Its Applications, Review Exercise 2.9
Review exercise 2.9. Answer the following questions for the vector space of 2 by 2 matrices: a) Does the set of 2 by 2 matrices with rank 1 form a subspace? b) What is the subspace spanned by the 2 … Continue reading
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Tagged invertible matrices, permutation matrices, positive matrices, spanning set, subspace
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