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Monthly Archives: August 2011
Linear Algebra and Its Applications, Exercise 2.2.10
Exercise 2.2.10. (a) Find all possible solutions to the following system: (b) What are the solutions if the right side is replaced with ? Answer: (a) Since the pivots of are in columns 1 and 3, the basic variables are … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.9
Exercise 2.2.9. Consider the following and : For what values of and does the system have a solution? Also, determine the nullspace of and provide two examples of vectors within it, and find the general solution to . Answer: We … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.8
Exercise 2.2.8. Consider the following system of linear equations: For what value of does this system have a solution? Answer: We use elimination to attempt to solve the system, starting by multiplying the first equation by 2 and subtracting it … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.7
Exercise 2.2.7. Consider the following system of linear equations: Find the values of for which the system has a solution. What is the rank? How many basic variables and free variables are there? Answer: We perform elimination by first subtracting … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.6
Exercise 2.2.6. Consider the following system of linear equations: Find the general solution expressed as the sum of a particular solution to and the solution to . Answer: We perform elimination by subtracting 2 times the first row from the … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.5
Exercise 2.2.5. Consider the system of linear equations represented by the following matrix: (This is the transpose of the matrix from exercise 2.2.4.) Find the echelon matrix , a set of basic variables, a set of free variables, and the … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.4
Exercise 2.2.4. Consider the system of linear equations represented by the following matrix: Find the echelon matrix , a set of basic variables, a set of free variables, and the general solution to . Then use elimination to find when … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.3
Exercise 2.2.3. Consider the system of linear equations represented by the following matrix: Find the factorization , a set of basic variables, a set of free variables, a general solution to (expressed as a linear combination as in equation (1) … Continue reading
Linear Algebra and Its Applications, Exercise 2.2.2
Exercise 2.2.2. Consider a system of linear equations that has more unknowns than there are equations and that has no solution. Provide the smallest example of such a system you can think of. Answer: Systems of one linear equation and … Continue reading
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Linear Algebra and Its Applications, Exercise 2.2.1
Exercise 2.2.1. Consider the set of 2 by 3 echelon matrices. How many patterns are there for such matrices? Answer: Suppose the first row of a 2 by 3 echelon matrix has a pivot in column 1. The second row … Continue reading
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