By Stephen Francis Andrilli; David Hecker
"Elementary Linear Algebra, Fourth version bargains computational options and primary theoretical effects vital to a primary path in linear algebra. The textual content makes a steady and gentle transition from computational effects related to vectors, matrices, and structures of linear equations to the final conception of summary vector areas. The textual content additionally presents a entire variety of functional functions, which will be coated in a versatile manner." "This textual content assists scholars to achieve a degree of summary pondering crucial for additional mathematical learn. The authors provide early, extensive recognition to the talents wanted for a scholar to develop into ok with the interpreting and writing of mathematical proofs. This emphasis is sustained in the course of the textual content with cautious statements of theorems and their proofs, and proof-related exercises."--BOOK JACKET. Read more...
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Additional info for Elementary linear algebra
If the scalar multiple of a vector is the zero vector, then either the scalar is zero or the vector is the zero vector. ■ ■ ■ Every vector in Rn is a linear combination of the standard unit vectors in Rn . The linear combinations of a given set of vectors represent the set of all possible “destinations” that can be reached using those vectors. Any vector v in R2 can be expressed as [||v|| cos , ||v|| sin ],where is the angle v forms with the positive x-axis. 14 CHAPTER 1 Vectors and Matrices ■ The resultant velocity of an object is the sum of its individual vector velocities.
8 Let x and y be nonzero vectors in Rn , and let be the angle between x and y. Then, (1) x · y > 0 if and only if 0 Յ < (2) x · y ϭ 0 if and only if ϭ (3) x · y < 0 if and only if radians (0◦ or acute). 2 radians (90◦ ). 2 < Յ radians (180◦ or obtuse). 2 The Dot Product 23 Special Cases: Orthogonal and Parallel Vectors aa Deﬁnition Two vectors x and y in Rn are orthogonal (perpendicular) if and only if x · y ϭ 0. Example 4 The vectors x ϭ [2, Ϫ5] and y ϭ [Ϫ10, Ϫ4] are orthogonal in R2 because x · y ϭ 0.
Deﬁnition and Properties of the Dot Product aa Deﬁnition Let x ϭ [x1 , x2 , . . , xn ] and y ϭ [y1 , y2 , . . The dot (inner) product of x and y is given by n x · y ϭ x1 y1 ϩ x2 y2 ϩ · · · ϩ xn yn ϭ xk yk . kϭ1 For example, if x ϭ [2, Ϫ4, 3] and y ϭ [1, 5, Ϫ2], then x · y ϭ (2)(1) ϩ (Ϫ4)(5) ϩ (3)(Ϫ2) ϭ Ϫ24. Notice that the dot product involves two vectors and the result is a scalar, whereas scalar multiplication involves a scalar and a vector and the result is a vector. Also, the dot product is not deﬁned for vectors having different numbers of coordinates.