# Elementary linear algebra by Stephen Francis Andrilli; David Hecker

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, whichRead more...

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**Example text**

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.