By Stewart Venit, Wayne Bishop, Jason Brown
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Additional resources for Elementary Linear Algebra, First Canadian Edition
Through P(2, -1) with normal n = (0, 2) 30 CHAPTER 1 r;e,_try ofJl2 ""J Jl3 11. Through P(- 2, 5) and perpendicular to the line x + 2y = 5 12. Through P(1, - 3) and perpendicular to the line 3x - 4y 1 In Exercises 13-21 give a point-parallel or two-point form, and para11etric equations for t~e 1 2 or 1 3 line described. 13. Through P(2, 1, - 3) and parallel to v 14. Through P(3, 1) and parallel to v (1, 2, 2) = (2, 3) 15. Through P(2, -3, 1) and parallel to the x>-axis 16. Through P(1, 2, -1) and Q(2, -1, 3) 17.
D+ kb, (4) which says that (sl' ~· ... ) satisfies Equation (3). Now assume that (sl' -'2• ... , s11) satisfies the ttansformed system. This implies that Equation (4) holds. We also know that Equation (1) is satisfied, so that ~s1 + ~-7. + · · · + a,s,. = b. s,) = kb. s,. d, 00 as desired. 2 Exercises In Exercises 1-6 determine isl1ear. J, 3-X:z - "3 - 4 x1 - x4 = 1- ~ x 1 I x4 I x 3 2 - 1 x+2xy-y-o - sin-X;z = x1 I - ~ I X:z - or not the ginn system of equations 2. 2x - \ly + X+ 2y- =0 3. 3x- xy 5.
If we start with three noncoplanar vectors u, v, and w, we have an analogous situation. The set of all linear combinations x = x(r, s, t) of u, v, and w given by x = ra I sv I tw is the 3-space determined by a, v, and w. Even if planes or 3-spaces do not contain the origin, a similar procedure can be used to obtain their equations: We first "translate" one of the points to the origin, cre-ating vectors with the same magnitude and direction as the directed line segments beginning at this point (Figure 4).
Elementary Linear Algebra, First Canadian Edition by Stewart Venit, Wayne Bishop, Jason Brown