Linear Algebra: Homework 3

Transcription

1 Linear Algebra: Homework 3 Alvin Lin August December 206 Section.2 Exercise 48 Find all values of the scalar k for which the two vectors are orthogonal. [ ] [ ] 2 k + u v 3 k u v 0 2(k + ) + 3(k ) 0 2k k 3 0 5k k 5 Exercise 49 Find all values of the scalar k for which the two vectors are orthogonal. k 2 u v k 2 3 u v 0 k 2 k 6 0 (k 3)(k + 2) 0 k 3 k 2 Exercise 50 Describe all vectors that are orthogonal to u [ ] 3. 3x + y 0 They are all parallel to the line described by y 3x.

2 Exercise 5 [ ] a Describe all vectors that are orthogonal to u. b They are all parallel to the line described by ax + by 0. Exercise 52 Under what conditions are the following true for vectors u and v in R 2 or R 3?. u + v u + v : This is true when the vectors are parallel. 2. u + v u v : This is true when the vectors are antiparallel. Exercise 53 Prove Theorem.2(b). Proof: u ( v + w) u v + u w u ( v + w) u i (v i + w i ) (u i v i + u i w i ) u i v i + u v + u w u i w i Exercise 54 Prove Theorem.2(d). Proof: u u 0 u u 0 iff u 0 u u u i u i (u i ) 2 (u i ) 2 is non-negative, therefore the summation must be greater than or equal to 0, and only equal to 0 when u 0. Exercise 55 Prove the stated property of distance between vectors. d( u, v) d( v, u) 2

3 Proof: d( u, v) n (u i v i ) 2 n ( ) 2 (v i u i ) 2 n (v i u i ) 2 d( v, u) Exercise 56 Prove the stated property of distance between vectors. d( u, w) d( u, v) + d( v, w) Proof: d( u, w) d( u, v) + d( v, w) u w u v + v w u w u v + v w (By the triangle inequality) u w u w Exercise 57 Prove the stated property of distance between vectors. d( u, v) 0 iff u v Proof: 0 d( u, v) 0 n (u i v i ) (u i v i ) 2 (u i v i ) u i 0 u v v u v i 3

4 Exercise 58 Prove that u c v c( u v). u c v c u i (cv i ) c(u i v i ) u i v i c( u v) Exercise 59 Prove that u v u v. Exercise 60 w + v w + v Let : w u v u v + v u v + v u u v + v u v u v u v u v (Triangle Inequality) Suppose know that u v u w. Does it follow that v w? If it does, give a proof that is valid in R n. Otherwise, give a counterexample. Suppose u 0. v and w can be any vector in R n. 0, 2 0 3, 4 Exercise 6 Prove that ( u + v) ( u v) u 2 v 2. ( u + v) ( u v) u u u v + v u v v u u v v u 2 v 2 Exercise 62a Prove that u + v 2 + u v 2 2 u v 2. u + v 2 + u v 2 ( u 2 + v 2 ) + ( u 2 + v 2 ) u 2 + v 2 + u 2 + v 2 2 u v 2 (Pythagorean Theorem) 4

5 Exercise 63 Prove that u v 4 u + v 2 4 u v 2. Exercise 64a 4 u + v 2 4 u v ( u + v) ( u + v) ( u v) ( u v) 4 4 ( u u + 2 u v + v v ( u u 2 u v + v v)) 4 u v Prove that u + v u v if and only if u and v are orthogonal. u + v u v ( u + v) ( u + v) ( u v) ( u v) ( u + v) ( u + v) ( u v) ( u v) u u + 2 u v + v v u u + v v 2 u v 0 u v 0 The dot product of two vectors is 0 if and only if the two vectors are orthogonal. Exercise 65a Prove that u + v and u v are orthogonal in R if and only if u v. Exercise 66 If u 2, v 3, and u v, find u + v. ( u + v) ( u v) 0 u u v v 0 u u v v u u v v u v u + v ( u + v) ( u + v) u u + 2 u v + v v u 2 + 2() + v

6 Exercise 67 Show that there are no vectors u and v such that u, v 2, and u v 3. u + v u + v u u v + v 2 u + v 2 + 2(3) Since this case violates the triangle inequality, there can be no such vectors. Exercise 68 Prove that if u is orthogonal to both v and w, then u is orthogonal to v + w. u v 0 u w 0 u v + u w 0 u ( v + w) 0 Prove that if u is orthogonal to both v and w, then u is orthogonal to s v + t w for all scalars s and t. Exercise 69 u v 0 u w 0 u v + u w 0 u ( v + w) 0 u i (v i + w i ) 0 u (v i + w i ) 0 Prove that u is orthogonal to v proj u v for all vectors u and v in R n, where u 0. u v u ( v proj u v) u ( v u u u) u v u v u u u u u v u v 0 6

7 Exercise 70a Prove that proj u (proj u v) proj u v. w proj u v u v u u u u v proj u (proj u v) proj u u u u proj u w u w u u u u v u u u u u u u u v u u u u u u u u v u u u proj u v Exercise 70b Prove that proj u ( v proj u v) 0. w proj u v u v u u u proj u ( v proj u v) proj u ( v w) u ( v w) u u u u v u w u u u u v u v u u u u u ( u u ) u v u v u u u u u u u u u ( ) u v u v u u u u u 0 u 0 7

8 Exercise 73 Use the fact that proj u v c u for some scalar c together with Figure.4 to find c and derive the formula for proj u v. Exercise 74 c u ( v c u) 0 (c u v) (c u c u) 0 c( u v) c 2 ( u u) u v u u c proj u v c u u v u u u Using mathematical induction, prove the following generalization of the Triangle Inequality: Basis: Assumption: Induction: v + v v n v + v v n u + v u + v v + v v n v + v v n v + v v n + v n+ v + v v n + v n+ v + v v n + v n+ Section.3 Exercise Write the equation of the line passing through P with normal vector n in normal form and general form. [ ] 3 P (0, 0) n 2 [ ] [ ] [ ] [ ] 3 x y 0 2 3x + 2y 0 Exercise 3 Write the equation of the line passing through P with direction vector d in vector form and parametric form. [ ] P (, 0) d 3 [ ] [ ] [ ] x + t y 0 3 { x t l y 3t 8

9 Exercise 5 Write the equation of the line passing through P with direction vector d in vector form and parametric form. P (0,, 0) d 4 x 0 y + t z 0 4 x t l y t z 4t Exercise 7 Write the equation of the line passing through P with normal vector n in normal form and general form. 3 P (0,, 0) n 2 3 x y 2 z 0 Exercise 9 3x + 2y + z 2 Write the equation of the plane passing through P with vectors u and v in vector form and parametric form. 2 3 P (0, 0, 0) u v 2 2 x y 0 + s + t 2 z 0 2 x 2s t3 Π y s + 2t z 2s + t Exercise Give the vector equation of the line passing through P and Q. P (, 2) Q (3, 0) [ ] 2 d 2 [ ] [ ] [ ] x t y 0 2 9

10 Exercise 3 Give the vector equation of the plane passing through P, Q, R. Exercise 5 P (,, ) Q (4, 0, 2) R (0,, ) 3 u v 0 2 x 3 y + s + t 0 z 2 Find parametric equations and an equation in vector form for the lines in R 2 with the following equations: Exercise 7 y 3x { x t l y 3t [ ] [ ] [ ] x 0 + t y 3 Suggest a vector proof of the fact that in R 2, two lines with slope m and m 2 are perpendicular if and only if m m 2. u m, v m 2, u v 0 m m m m 2 0

11 Exercise 9 The plane P has the equation 4x y +5z 2. For each of the planes P in Exercise 8. determine whether P and P are parallel, perpendicular, or neither. a : 2x + 3y z n 4,, 5 n a 2, 3, n c n a n n a 8 + ( 3) + ( 5) 0 b : 4x y + 5z 0 n b 4,, 5 n c n a c : x y z 3 n c,, n c n c n n c ( 5) 0 d : 4x + 6y 2z 0 n d 4, 6, 2 2, 3, n c n d n n d 8 + ( 3) + ( 5) 0 Perpendicular Parallel Perpendicular Perpendicular Exercise 2 Find the vector form of the equation of the line in R 2 that passes through P (2, ) and is parallel to the line with general equation 2x 3y. d 2, 3 Exercise 23 [ ] x y [ 2 ] + t Find the vector form of the equation of the line in R 3 that passes through P (, 0, 3) and is parallel to the line with parametric equations: x t l y 2 + 3t z 2 t x y 0 + t 3 z 3 Section 2. Exercise Determine which equations are linear in the variables x, y, z. [ ] 2 3 x πy + 3 5z 0

12 Linear. Exercise 3 Determine which equations are linear in the variables x, y, z. Nonlinear, x is not st degree. x + 7y + z sin π 9 Exercise 5 Nonlinear, cos(x) is a function of x. 3 cos(x) 4y + z 3 Exercise Find the solution set of each equation. 3x 6y 0 All points that lie on the line y 2 x. Exercise 3 Find the solution set of each equation. x + 2y + 3z 4 All points that lie on the plane x + 2y + 3z 4 or [4 3t 2s, s, t]. Exercise 5 Determine geometrically whether each system has a unique solution, infinitely many solutions, or no solution. Then solve each system algebraically to confirm your answer. Two intersecting lines, one solution at (3,-3). x + y 0 2x + y 3 Exercise 7 Determine geometrically whether each system has a unique solution, infinitely many solutions, or no solution. Then solve each system algebraically to confirm your answer. Two parallel lines, no solution. 3x 6y 3 x + 2y Exercise 9 Solve the given system by back substitution. x 2y y 3 x 7 2

13 Exercise 2 Solve the given system by back substitution. x y + z 0 2y z 3z z 3 y 3 x 2 3 Exercise 23 Solve the given system by back substitution. Exercise 25 Solve these systems. Exercise 27 Find the augmented matrices of the linear systems. x + x 2 x 3 x 4 x 2 + x 3 + x 4 0 x 3 x 4 0 x 4 x 3 x 2 2 x x 2 2x + y 3 3x 4y + z x y 0 y 7 z 23 2x + y 3 [ ] 0 A 2 3 3

14 Exercise 29 Find the augmented matrices of the linear systems. Exercise 3 x + 5y x + y 5 2x + 4y 4 A Find a system of linear equations that has the given matrix as its augmented matrix. 0 A 0 2 Exercise 33 y + z x y 2x y + z Solve the linear systems in the given equations. x y 0 2x + y 3 [ ] 0 A 2 3 2R + R 2 R 2 [ ] 0 A R 2 R 2 [ ] 0 A 0 R 2 + R R [ ] 0 A 0 x y 4

15 Exercise 35 Solve the linear systems in the given equations. x + 5y x + y 5 2x + 4y 4 A R + R 2 R R + R 3 R 3 5 A R 2 R 2 6 R 3 R 3 5 A 0 0 5R 3 + R R 0 4 A 0 0 x 4 y 5

16 Exercise 37 Solve the linear systems in the given equations. Exercise 39 A R 2 R R + R 3 R R 3 R 2 R (Inconsistent) Find a system of two linear equations in the variables x and y whose solution set is given by the parametric equations x t and y 3 2t. y 3 2x 3x 5 Find another parametric solution to the system in the first part in which the parameter is s and y s. y s x s 3 2 If you have any questions, comments, or concerns, please contact me at alvin@omgimanerd.tech 6