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With conventional linear algebra texts, the path is comparatively effortless for college kids in the course of the early levels as fabric is gifted in a well-recognized, concrete atmosphere. even though, while summary strategies are brought, scholars usually hit a wall. teachers appear to agree that yes suggestions (such as linear independence, spanning, subspace, vector house, and linear changes) aren't simply understood and require time to assimilate. those ideas are basic to the examine of linear algebra, so scholars' figuring out of them is essential to getting to know the topic. This textual content makes those recommendations extra available through introducing them early in a well-known, concrete Rn atmosphere, constructing them progressively, and returning to them during the textual content in order that after they are mentioned within the summary, scholars are conveniently in a position to understand.
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Extra resources for Linear Algebra and Its Applications (5th Edition)
In Exercises 21 and 22, mark each statement True or False. Justify each answer. four 6. Repeat Exercise 5 for a nonzero three 2 matrix. Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14. Ä Ä 1 three four 7 1 three zero five 7. eight. three nine 7 6 three 7 zero nine Ä Ä zero 1 2 three 1 2 1 four 10. nine. 1 three four 6 2 four five 6 three 2 three 2 1 zero nine zero four three 2 four zero 60 1 three zero 17 7 6 12 05 12. 6 eleven. four nine forty zero zero 1 seventy five 6 four eight zero zero zero zero zero 1 three 2 1 three zero 1 zero 2 7 60 1 zero zero four 1 7 thirteen. 6 forty zero zero 1 nine forty five zero zero zero zero zero zero three 2 1 zero five zero eight three 60 1 four 1 zero sixty seven 7 14. 6 forty zero zero zero 1 05 zero zero zero zero zero zero Exercises 15 and 16 use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique. 2 three 15. a. four zero zero 2 zero b. four zero zero zero zero zero zero zero zero five three zero five 19. x1 C hx2 D 2 4x1 C 8x2 D okay 20. x1 3x2 D 1 2x1 C hx2 D ok 21. a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. b. The row reduction algorithm applies only to augmented matrices for a linear system. c. A uncomplicated variable in a linear procedure is a variable that corresponds to a pivot column in the coefﬁcient matrix. d. Finding a parametric description of the solution set of a linear system is the same as fixing the system. e. If one row in an echelon form of an augmented matrix is Œ zero zero zero five zero , then the associated linear system is inconsistent. 22. a. The reduced echelon form of a matrix is unique. b. If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. c. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. d. A general solution of a system is an explicit description of all solutions of the system. e. Whenever a system has free variables, the solution set contains many solutions. 23. Suppose the coefﬁcient matrix of a linear system of four equations in four variables has a pivot in each column. Explain why the system has a unique solution. 24. Suppose a system of linear equations has a three five augmented matrix whose ﬁfth column is not a pivot column. Is the system consistent? Why (or why not)? True/false questions of this type will appear in many sections. tools for justifying your answers were described before Exercises 23 and 24 in part 1. 1. four 1. 2 25. Suppose the coefﬁcient matrix of a system of linear equations has a pivot position in every row. Explain why the system is constant. 26. Suppose a three five coefﬁcient matrix for a system has three pivot columns. Is the system consistent? Why or why not? 27. Restate the last sentence in Theorem 2 using the concept of pivot columns: “If a linear system is consistent, then the solution is unique if and only if . ” 28. What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution? 29. A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system.