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Chapter 4: Vector Spaces

Expert-verified
Linear Algebra and its Applications
Pages: 191 - 266
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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Illustration

229 Questions for Chapter 4: Vector Spaces

  1. Question 10: Determine if \(P = \left[ {\begin{array}{*{20}{c}}1&{.2}\\0&{.8}\end{array}} \right]\) is a regular stochastic matrix.

    Found on Page 191

  2. If the null space of A \({\bf{7}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A?

    Found on Page 191

  3. In Exercises 7-10, let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2}} \right\}\) and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},{{\mathop{\rm c}\nolimits} _2}} \right\}\) be bases for \({\mathbb{R}^2}\). In each exercise, find the change-of-coordinates matrix from \(B\) to \(C\) and the change-of-coordinates matrix from \(C\) to \(B\).

    Found on Page 191

  4. In Exercies 7-12, assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems.

    Found on Page 191

  5. In Exercises 9 and 10, find the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^n}\).

    Found on Page 191

  6. Let S be a maximal linearly independent subset of a vector space V. In other words, S has the property that if a vector not in S is adjoined to S, the new set will no longer be linearly independent. Prove that S must be a basis of V. [Hint: What if S were linearly independent but not a basis of V?]

    Found on Page 191

  7. If the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?

    Found on Page 191

  8. Find a basis for the set of vectors in\({\mathbb{R}^{\bf{3}}}\)in the plane\(x + {\bf{2}}y + z = {\bf{0}}\). (Hint:Think of the equation as a “system” of homogeneous equations.)

    Found on Page 191

  9. In Exercises 11 and 12, and Care bases for a vector space V. Mark each statement True or False. Justify each answer.\(B\)

    Found on Page 191

  10. In Exercises 11 and 12, use an inverse matrix to find for the given \({\mathop{\rm x}\nolimits} \) and \(B\).

    Found on Page 191

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