By T. S. Blyth, E. F. Robertson
Problem-solving is an artwork significant to knowing and talent in arithmetic. With this sequence of books, the authors have supplied a variety of labored examples, issues of whole recommendations and try papers designed for use with or rather than regular textbooks on algebra. For the benefit of the reader, a key explaining how the current books can be utilized at the side of many of the significant textbooks is incorporated. every one quantity is split into sections that commence with a few notes on notation and conditions. the vast majority of the fabric is aimed toward the scholars of usual skill yet a few sections include tougher difficulties. through operating during the books, the scholar will achieve a deeper knowing of the elemental recommendations concerned, and perform within the formula, and so answer, of alternative difficulties. Books later within the sequence disguise fabric at a extra complicated point than the sooner titles, even though each one is, inside its personal limits, self-contained.
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Extra resources for Algebra Through Practice: A Collection of Problems in Algebra with Solutions
Deﬁnition 1 A solution to (1) is a set of n scalars x1 , x2 , . . , xn that when substituted into (1) satisﬁes the given equations (that is, the equalities are valid). System (1) is a generalization of systems considered earlier in that m can differ from n. If m > n, the system has more equations than unknowns. If m < n, the system has more unknowns than equations. If m = n, the system has as many unknowns as equations. 3 may be used to convert (1) into the matrix form Ax = b, (2) 43 44 Chapter 2 where Simultaneous Linear Equations ⎡ ⎤ a1n a2n ⎥ ⎥ ..
Prove that if a 2 × 2 matrix A commutes with every 2 × 2 diagonal matrix, then A must also be diagonal. Hint: Consider, in particular, the diagonal matrix D= 1 0 0 . 0 18. Prove that if an n × n matrix A commutes with every n × n diagonal matrix, then A must also be diagonal. 19. Compute D2 and D3 for the matrix D deﬁned in Problem 15. 20. Find A3 if ⎡ 1 A = ⎣0 0 0 2 0 ⎤ 0 0⎦. 3 21. Using the results of Problems 19 and 20 as a guide, what can be said about Dn if D is a diagonal matrix and n is a positive integer?
Example 1 Graph the vectors v = [2 magnitude and angle of each. 5. 4◦ . 1 = −1, and −1 θ = 135◦ . 14, tan θ = To construct the sum of two vectors u + v geometrically, graph u normally, translate v so that its initial point coincides with the terminal point of u, being careful to preserve both the magnitude and direction of v, and then draw an arrow from the origin to the terminal point of v after translation. 6 represents the sum u + v. 6 for the two vectors deﬁned in Example 1. To construct the difference of two vectors u − v geometrically, graph both u and v normally and construct an arrow from the terminal point of v to the terminal point of u.