1Linear  Algebra Problem  Set  1.1~2.2

1.1 Vectors and Linear Combinations (4, 11, 13, 26, 28)

4. From 21

v ⎡ ⎤= ⎢ ⎥⎣ ⎦

and 12

w ⎡ ⎤= ⎢ ⎥⎣ ⎦

, find the components of 3v+w and v-3w and cv+dw.

Sol. 2 1 7 2 1 1 2 1 2

3 3 , 3 3 ,1 2 5 1 2 5 1 2 2

c dv w v w cv dw c d

c d− +⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

+ = + = − = − = + = + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 11. Four corners of the cube are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1). What are the other four corners? Find the coordinates of the center point of the cube. The center points of the six faces are . Sol. (a) (0, 1, 1), (1, 0, 1), (1, 1, 1), (1, 1, 0)

(b) 1 1 1( , , )2 2 2

(c) 1 1 1 1 1 1 1 1 1 1 1 1( , ,1), ( , ,0), ( ,0, ), ( ,1, ), (0, , ), (1, , )2 2 2 2 2 2 2 2 2 2 2 2

13. (a) What is the sum V of the twelve vectors that go from the center of a clock to the

hours 1:00, 2:00, …, 12:00? (b) If the vector to 4:00 is removed, find the sum of the eleven remaining vectors. (c) What is the unit vector to 1:00? Sol. (a) 0 (b) -4:00

(c) 1 3(cos ,sin ) ( , )3 3 2 2π π

=

26. What combination of the vectors 12⎡ ⎤⎢ ⎥⎣ ⎦

and 31⎡ ⎤⎢ ⎥⎣ ⎦

produces 148

⎡ ⎤⎢ ⎥⎣ ⎦

? Express this

question as two equations for the coefficients c and d in the linear combination. Sol.

1 3 142 1 8

c d⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ →

3 142 8c d

c d+ =⎧

⎨ + =⎩ → c=2, d=4

28. If (a, b) is a multiple of (c, d) with abcd≠0, show that (a, c) is a multiple of (b, d). Sol.

a=kc, b=kd, (a, c) = (kc, c), (b, d) = (kd, d), (b, d) = (kd, d) × cd→ (kc, c)

 2Linear  Algebra Problem  Set  1.1~2.2

1.2 Lengths and Dot Products (3, 8, 17, 23, 31) 3. Find unit vectors in the directions of v and w, and the cosine of the angle θ. Choose vectors that make 0° , 90° , and 180° angles with w.

3 4,

4 3v w⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Sol.

(a) unit vector 2 2

(3, 4) 3 4( , )5 53 4

vv= =

+,

2 2

(4,3) 4 3( , )5 54 3

ww

= =+

(b) 24cos25

v wv w

θ ⋅= =

(c) w, u, -w 8. True or false (give a reason if true or a counterexample if false): (a) If u is perpendicular (in three dimensions) to v and w, then v and w are parallel. (b) If u is perpendicular to v and w, then u is perpendicular to v+2w. (c) If u and v are perpendicular unit vectors then 2u v− = .

Sol. (a) F, counterexample: u=(1, 0, 0), v=(0, 1, 0), w = (0, 1, 1) (b) T, u⋅(v+2w)=u⋅v+2u⋅w=0+0=0

(c) T, 2 2 2 2( ) ( ) 2 2u v u v u v u uv v u v− = − ⋅ − = − + = + =

17. What are the cosines of the angles α, β, θ between the vector (1, 0, -1) and the unit vectors i, j, k along the axes? Check the formula cos2α + cos2β+ cos2θ =1. Sol.

(a)

(1,0, 1) (1,0,0) 1 (1,0, 1) (0,1,0)cos ,cos 0,(1,0, 1) (1,0,0) (1,0, 1) (0,1,0)2(1,0, 1) (0,0,1) 1cos(1,0, 1) (0,0,1) 2

α β

θ

− ⋅ − ⋅= = = =

− −

− ⋅= =

−

(b) 2 2 2 1 1cos cos cos 0 12 2

α β θ+ + = + + =

23. The figure shows that cosα = v1/||v|| and sinα = v2/||v||. Similarly cosβ is and sinβ is . The angle θ is β-α. Substitute into the formula cosβcosα + sinβsinα for cos(β-α) to find cosθ = v⋅w/||v|| ||w||. Sol. (a) cosβ = w1/||w|| (b) sinβ = w2/||w

 3Linear  Algebra Problem  Set  1.1~2.2

(c) 1 1 2 2 1 1 2 2cos cos sin sin|| || || || || || || || || || || ||w v w v w v w vw v w v w v

β α β α ++ = + =

31. Can three vectors in the xy plane have u⋅v<0, and v⋅w<0 and u⋅w<0? Sol.

1 1 1, ,

0 2 1u v w

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

 4Linear  Algebra Problem  Set  1.1~2.2

2.1 Vectors and Linear equations (4, 10, 17, 19, 34) 4. If equation 1 is added to equation 2, which of these are changed: the planes in the row picture, the column picture, the coefficient matrix, the solution?

1 0 0 2 1 0 0 20 1 0 3 1 1 0 50 0 1 4 0 0 1 4

x y z x y zx y z x y zx y z x y z

+ + = + + =⎧ ⎧⎪ ⎪+ + = ⇒ + + =⎨ ⎨⎪ ⎪+ + = + + =⎩ ⎩

Sol. Changed: the planes in the row picture, the column picture, the coefficient matrix Unchanged: the solution

10. Compute each Ax by dot products of the rows with the column vector.

(a) 1 2 4 22 3 1 24 1 2 3

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

(b)

2 1 0 0 11 2 1 0 10 1 2 1 10 0 1 2 2

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Sol.

(a) 1 2 4 2 182 3 1 2 54 1 2 3 0

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(b)

2 1 0 0 1 31 2 1 0 1 40 1 2 1 1 50 0 1 2 2 5

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

17.

(a) What 2 by 2 matrix R rotates every vector by 90° ? R times xy⎡ ⎤⎢ ⎥⎣ ⎦

is yx

⎡ ⎤⎢ ⎥−⎣ ⎦

(b) What 2 by 2 matrix R rotates every vector by 180° ? Sol.

(a) 0 11 0

⎡ ⎤⎢ ⎥−⎣ ⎦

(b) 1 0

0 1−⎡ ⎤⎢ ⎥−⎣ ⎦

19. What 2 by 2 matrix E subtracts the first component from the second component? What 3 by 3 matrix does the same?

 5Linear  Algebra Problem  Set  1.1~2.2

3 35 2

E ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ and

3 35 27 7

E⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Sol.

(a) 1 01 1

E ⎡ ⎤= ⎢ ⎥−⎣ ⎦

(b) 1 0 01 1 0

0 0 1E

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

34. Invent a 3 by 3 magic matrix M3 with entries 1, 2, …, 9. All rows and columns and diagonals add to 15. The first row could be 8, 3, 4. What is M3 times (1, 1, 1)? What is M4 times (1, 1, 1, 1) if this magic matrix has entries 1, …, 16? Sol.

(a) 3

5 5 5 8 3 4 8 3 4 1 155 5 5 1 5 9 , 1 5 9 1 15

5 5 5 6 7 2 6 7 2 1 15

u v u vM u v u v

v v u u

+ + − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − + + = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ − + −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(b) (1 16) 16 1 342 4

+ ×× = , (34, 34, 34, 34)

 6Linear  Algebra Problem  Set  1.1~2.2

2.2 The Idea of Elimination (5, 6, 7, 12, 21, 27) 5. Choose a right side which gives no solution and another right side which gives infinitely many solutions. What are two of those solutions?

3 2 106 4

x yx y+ =+ =

Sol. No solution: ≠ 20 Infinitely many solutions: 20, (0, 5), (2, 2) 6. Choose a coefficient b that makes this system singular. Then choose a right side g that makes it solvable. Find two solutions in that singular case.

2 164 8

x byx y g+ =+ =

Sol. b=4, g=32, (8, 0), (0, 4) 7. For which numbers a does elimination break down (1) permanently (2) temporarily?

3 34 6 6ax y

x y+ = −+ =

Solve for x and y after fixing the second breakdown by a row exchange. Sol. (1) a =2 (2) a =0, (3, -1) 12. Apply elimination (circle the pivots) and back substitution to solve

2 3 34 5 72 3 5

x yx y zx y z

− =− + =− − =

List the three row operations: Subtract times row from row . Sol.

2 3 0 3 2 3 0 3 2 3 0 34 5 1 7 0 1 1 1 0 1 1 12 1 3 5 0 2 3 2 0 0 5 0

⎡ − ⎤ ⎡ − ⎤ ⎡ − ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− → →⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x=3, y=1, z=0 Subtract 2 times row 1 from row 2 .

 7Linear  Algebra Problem  Set  1.1~2.2

Subtract 1 times row 1 from row 3 . Subtract 2 times row 2 from row 3 .

21. Find the pivots and the solution for these four equations:

2 02 0

2 02 5

x yx y z

y z tz t

+ =+ + =

+ + =+ =

Sol.

2 1 0 0 0 2 1 0 0 0 2 1 0 0 0 2 1 0 0 01 2 1 0 0 0 3/ 2 1 0 0 0 3/ 2 1 0 0 0 3/ 2 1 0 00 1 2 1 0 0 1 2 1 0 0 0 4 / 3 1 0 0 0 4 / 3 1 00 0 1 2 5 0 0 1 2 5 0 0 1 2 5 0 0 0 5 / 4 5

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥→ → →⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x=-1, y=2, z=-3, t=4 27. Look for a matrix that has row sums 4 and 8, and column sums 2 and s:

a 4 2Matrix

8b a b a c

c d c d b d s+ = + =⎡ ⎤

= ⎢ ⎥ + = + =⎣ ⎦

The four equations are solvable only if s= . Sol. s=10