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Chapter 1: Linear Equations in Linear Algebra

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Linear Algebra and its Applications
Pages: 1 - 92
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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410 Questions for Chapter 1: Linear Equations in Linear Algebra

  1. In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation.

    Found on Page 1

  2. In Exercise 1-10, assume that \(T\) is a linear transformation. Find the standard matrix of \(T\).

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  3. In Exercises 9 and 10, (a) for what values of h is \({{\bf{v}}_3}\) inSpan \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2}} \right\}\), and (b) for what values of \(h\) is \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) linearlydependent? Justify each answer.

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  4. In a certain region, about 6% of city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where\({{\bf{x}}_{\bf{0}}}\)is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.

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  5. In Exercises 7-12, describe all solutions of \(Ax = 0\) in parametric vector form, where \(A\) is row equivalent to the given matrix.

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  6. In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

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  7. Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

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  8. In Exercises 10, write a vector equation that is equivalent tothe given system of equations.

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  9. For Exercises 9 and 10, find all x in \({\mathbb{R}^{\bf{4}}}\) that are mapped into the zero vector by the transformation \({\bf{x}}| \to A{\bf{x}}\) for the given matrix A.

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  10. Balance the chemical equations in Exercise 5-10 using the vector equation approach discussed in this section.

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