29 62 CHAPTER 1 Linear Equations in Linear Algebra9. V₁ =10. V₁ =11.13.-[-]---3215.In Exercises 11-14, find the value(s) of h for which the vectorsare linearly dependent. Justify each answer.17.-----=10-3---5-353-57-819. 12-4-5BED AND5-97685h1.[8]096-65439Determine by inspection whether the vectors in Exercises 15-20are linearly independent. Justify each answer.[{][B][B][-]112.14.16.2-9h2HO-47-318.6[][]-29220.4 0₁In Exercises 21 and 22, mark each statement True or False. Justifyeach answer on the basis of a careful reading of the text.h[4][3][³][8]5-7--2530[8]21. a. The columns of a matrix A are linearly independent if theequation Ax = 0 has the trivial solution.b. If S is a linearly dependent set, then each vector is a linearcombination of the other vectors in S.c. The columns of any 4 x 5 matrix are linearly dependent.d. If x and y are linearly independent, and if {x, y, z) islinearly dependent, then z is in Span {x,y}.22. /a. Two vectors are linearly dependent if and only if they lieon a line through the origin.b. If a set contains fewer vectors than there are entries in thevectors, then the set is linearly independent.c. If x and y are linearly independent, and if z is inSpan {x, y), then {x, y, z) is linearly dependent.d. If a set in R" is linearly dependent, then the set containsmore vectors than there are entries in each vector.In Exercises 23-26, describe the possible echelon forms of thematrix. Use the notation of Example 1 in Section 1.2.23. A is a 3 x 3 matrix with linearly independent columns.24. A is a 2 x 2 matrix with linearly dependent columns,25. A is a 4 × 2 matrix, A = [a, a₂], and a2 is not aal.a₂}26. A is a 4 x 3 matrix, A = [a, a2 a3], such that {a,, a2) islinearly independent and a3 is not in Span {a,, a2}.27. How many pivot columns must a 7 x 5 matrix have if itscolumns are linearly independent? Why?28. How many pivot columns must a 5 x 7 matrix have if itscolumns span R5? Why?29. Construct 3 x 2 matrices A and B such that Ax = 0 has onlythe trivial solution and Bx = 0 has a nontrivial solution.30. a. Fill in the blank in the following statement: "If A isan m x n matrix, then the columns of A are linearlyindependent if and only if A has pivot columns."b. Explain why the statement in (a) is true.Exercises 31 and 32 should be solved without performing rowoperations. [Hint: Write Ax = 0 as a vector equation.]31. Given A =multiple of2 3 532. Given A =-5 1-3 -11 0-4-414-7is the sum of the first two columns. Find a nontrivial solutionof Ax = 0.1659 -3observe that the third column3 , observe that the first column3plus twice the second column equals the third column. Finda nontrivial solution of Ax = 0.Each statement in Exercises 33-38 is either true (in all cases)or false (for at least one example). If false, construct a specificexample to show that the statement is not always true. Such anexample is called a counterexample to the statement. If a statementis true, give a justification. (One specific example cannot explainwhy a statement is always true. You will have to do more workhere than in Exercises 21 and 22.)33. If v₁,..., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4)is linearly dependent.34. If V₁,..., V4 are in R4 and v3 = 0, then {V₁, V2, V3, V4) islinearly dependent.35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of V₁.then {V₁, V₂} is linearly independent.36. If V₁,..., V4 are in R4 and v3 is not a linear combination ofV₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent.37. If V₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent.then {V1, V2, V3, V4) is also linearly dependent.38. If V₁,..., V4 are linearly independent vectors in R4, then{V1, V2, V3} is also linearly independent. [Hint: Think aboutXIVI + X₂V₂ + X3V3 +0 V4 = 0.]1.

Solution to the given question, Introduction: The solution of a matrix equation AX=0 is a trivial…

Answered: 29. Construct 3 x 2 matrices A and B…

29. Construct 3 x 2 matrices A and B such that Ax = 0 has only the trivial solution and Bx = 0 has a nontrivial solution.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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