Linear Algebra Problem Set 1.1~2.2
1.1 Vectors and Linear Combinations (4, 11, 13, 26, 28)
⎡ 2⎤
⎡1 ⎤
4. From v = ⎢ ⎥ and w = ⎢ ⎥ , find the components of 3v+w and v-3w and cv+dw.
⎣1 ⎦
⎣2⎦
Sol.
⎡ 2 ⎤ ⎡1 ⎤ ⎡7 ⎤
⎡ 2 ⎤ ⎡1 ⎤ ⎡ −1⎤
⎡2⎤
⎡1 ⎤ ⎡ 2c + d ⎤
3v + w = 3 ⎢ ⎥ + ⎢ ⎥ = ⎢ ⎥ , v − 3w = ⎢ ⎥ − 3 ⎢ ⎥ = ⎢ ⎥ , cv + dw = c ⎢ ⎥ + d ⎢ ⎥ = ⎢
⎥
⎣1 ⎦ ⎣ 2 ⎦ ⎣5 ⎦
⎣ 1 ⎦ ⎣ 2 ⎦ ⎣ −5 ⎦
⎣1 ⎦
⎣ 2 ⎦ ⎣ c + 2d ⎦
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)
1 1 1
(b) ( , , )
2 2 2
1 1
1 1
1 1 1 1
1 1
1 1
(c) ( , ,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) (cos
π
π
1 3
,sin ) = ( , )
3
3
2 2
⎡1 ⎤
⎡ 3⎤
⎡14 ⎤
26. What combination of the vectors ⎢ ⎥ and ⎢ ⎥ produces ⎢ ⎥ ? Express this
⎣ 2⎦
⎣1⎦
⎣8⎦
question as two equations for the coefficients c and d in the linear combination.
Sol.
⎡1 ⎤
⎡3⎤ ⎡14 ⎤
c⎢ ⎥+d ⎢ ⎥ = ⎢ ⎥ →
⎣2⎦
⎣1⎦ ⎣ 8 ⎦
⎧c + 3d = 14
⎨
⎩ 2c + d = 8
→ 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) ×
c
→ (kc, c)
d
1
Linear 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⎤
v = ⎢ ⎥,w = ⎢ ⎥
⎣4⎦
⎣3⎦
Sol.
(3, 4)
3 4
v
=
= ( , ),
5 5
v
32 + 42
v⋅w
24
(b) cos θ =
=
v w 25
(a) unit vector
(4,3)
4 3
w
=
=( , )
5 5
w
42 + 32
(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 u − v = 2 .
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, u − v = (u − v) ⋅ (u − v) = u 2 − 2uv + v 2 = u 2 + v 2 = 2
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.
(1, 0, −1) ⋅ (1, 0, 0)
1
(1, 0, −1) ⋅ (0,1, 0)
cos α =
=
, cos β =
= 0,
(1, 0, −1) (1, 0, 0)
(1, 0, −1) (0,1, 0)
2
(a)
(1, 0, −1) ⋅ (0, 0,1)
1
cos θ =
=
(1, 0, −1) (0, 0,1)
2
1
1
(b) cos 2 α + cos 2 β + cos 2 θ = + 0 + = 1
2
2
and
23. The figure shows that cosα = v1/||v|| and sinα = v2/||v||. Similarly cosβ is
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
2
Linear Algebra Problem Set 1.1~2.2
(c) cos β cos α + sin β sin α =
w1 v1
w v2
w v + w2 v2
+ 2
= 11
|| w || || v || || w || || v || || w || || v ||
31. Can three vectors in the xy plane have u⋅v<0, and v⋅w<0 and u⋅w<0?
Sol.
⎡1 ⎤
⎡ −1⎤
⎡ −1⎤
u = ⎢ ⎥,v = ⎢ ⎥,w = ⎢ ⎥
⎣0⎦
⎣2⎦
⎣ −1⎦
3
Linear 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?
⎧1x + 0 y + 0 z = 2 ⎧1x + 0 y + 0 z = 2
⎪
⎪
⎨ 0 x + 1 y + 0 z = 3 ⇒ ⎨1x + 1 y + 0 z = 5
⎪0 x + 0 y + 1z = 4 ⎪0 x + 0 y + 1z = 4
⎩
⎩
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.
⎡ 2 1 0 0 ⎤ ⎡1 ⎤
⎡ 1 2 4⎤ ⎡2⎤
⎢1 2 1 0 ⎥ ⎢1 ⎥
⎥⎢ ⎥
(a) ⎢⎢ −2 3 1 ⎥⎥ ⎢⎢ 2 ⎥⎥
(b) ⎢
⎢0 1 2 1 ⎥ ⎢1 ⎥
⎢⎣ −4 1 2 ⎥⎦ ⎢⎣ 3 ⎥⎦
⎢
⎥⎢ ⎥
⎣0 0 1 2⎦ ⎣2⎦
Sol.
⎡ 1 2 4 ⎤ ⎡ 2 ⎤ ⎡18⎤
(a) ⎢⎢ −2 3 1 ⎥⎥ ⎢⎢ 2 ⎥⎥ = ⎢⎢ 5 ⎥⎥
⎢⎣ −4 1 2 ⎥⎦ ⎢⎣ 3 ⎥⎦ ⎢⎣ 0 ⎥⎦
⎡2
⎢1
(b) ⎢
⎢0
⎢
⎣0
1
2
1
0
0
1
2
1
0 ⎤ ⎡1 ⎤ ⎡ 3⎤
0 ⎥⎥ ⎢⎢ 1 ⎥⎥ ⎢⎢ 4 ⎥⎥
=
1 ⎥ ⎢1 ⎥ ⎢5 ⎥
⎥⎢ ⎥ ⎢ ⎥
2⎦ ⎣ 2⎦ ⎣5 ⎦
17.
⎡x⎤
⎡y⎤
(a) What 2 by 2 matrix R rotates every vector by 90° ? R times ⎢ ⎥ is ⎢ ⎥
⎣− x ⎦
⎣ y⎦
(b) What 2 by 2 matrix R rotates every vector by 180° ?
Sol.
⎡ 0 1⎤
(a) ⎢
⎥
⎣ −1 0 ⎦
⎡ −1 0 ⎤
(b) ⎢
⎥
⎣ 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?
4
Linear Algebra Problem Set 1.1~2.2
⎡3⎤ ⎡3⎤
⎡ 3⎤ ⎡ 3 ⎤
E ⎢ ⎥ = ⎢ ⎥ and E ⎢⎢ 5 ⎥⎥ = ⎢⎢ 2 ⎥⎥
⎣5 ⎦ ⎣ 2 ⎦
⎢⎣7 ⎥⎦ ⎢⎣7 ⎥⎦
Sol.
⎡ 1 0⎤
(a) E = ⎢
⎥
⎣ −1 1 ⎦
⎡ 1 0 0⎤
(b) E = ⎢⎢ −1 1 0 ⎥⎥
⎢⎣ 0 0 1 ⎥⎦
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.
5+v −u
5 − v ⎤ ⎡8 3 4 ⎤
⎡ 5+u
⎢
5
5 + u + v ⎥⎥ = ⎢⎢1 5 9 ⎥⎥ ,
(a) M 3 = ⎢5 − u − v
⎢⎣ 5 + v
5−v+u
5 − u ⎥⎦ ⎢⎣ 6 7 2 ⎥⎦
(b)
(1 + 16) ×16 1
× = 34 , (34, 34, 34, 34)
2
4
⎡8 3 4 ⎤ ⎡1⎤ ⎡15⎤
⎢1 5 9 ⎥ ⎢1⎥ = ⎢15⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢⎣6 7 2 ⎥⎦ ⎢⎣1⎥⎦ ⎢⎣15⎥⎦
5
Linear 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?
3x + 2 y = 10
6x + 4 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 x + by = 16
4x + 8 y = g
Sol.
b=4, g=32, (8, 0), (0, 4)
7. For which numbers a does elimination break down (1) permanently (2)
temporarily?
ax + 3 y = −3
4x + 6 y = 6
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
2x − 3y
=3
4x − 5 y + z = 7
2 x − y − 3z = 5
List the three row operations: Subtract
row
.
times row
Sol.
⎡ 2 −3 0 3 ⎤ ⎡ 2 −3 0 3 ⎤ ⎡ 2 −3 0 3⎤
⎢
⎥ ⎢
⎥ ⎢
⎥
⎢ 4 −5 1 7 ⎥ → ⎢ 0 1 1 1 ⎥ → ⎢ 0 1 1 1 ⎥
⎢⎣ 2 −1 −3 5 ⎥⎦ ⎢⎣ 0 2 −3 2 ⎥⎦ ⎢⎣ 0 0 −5 0 ⎥⎦
x=3, y=1, z=0
c Subtract
2
times row
1
from row
2
.
from
6
Linear Algebra Problem Set 1.1~2.2
7
d Subtract
e Subtract
1
2
times row
times row
1
2
from row
from row
3
3
.
.
21. Find the pivots and the solution for these four equations:
2x + y
=0
x + 2y + z
=0
y + 2z + t = 0
z + 2t = 5
Sol.
⎡2
⎢
⎢1
⎢0
⎢
⎢⎣ 0
1 0 0 0⎤ ⎡2 1
⎥ ⎢
2 1 0 0⎥ ⎢0 3 / 2
→
1 2 1 0⎥ ⎢0 1
⎥ ⎢
0 1 2 5 ⎥⎦ ⎢⎣ 0 0
0 0 0⎤ ⎡2 1
0
⎥ ⎢
1 0 0⎥ ⎢0 3 / 2 1
→
2 1 0⎥ ⎢0 0 4 / 3
⎥ ⎢
1 2 5 ⎥⎦ ⎢⎣ 0 0
1
0 0⎤ ⎡2 1
0
0 0⎤
⎥ ⎢
⎥
0 0⎥ ⎢0 3 / 2 1
0 0⎥
→
1 0⎥ ⎢0 0 4 / 3 1 0⎥
⎥ ⎢
⎥
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 b ⎤ a + b = 4 a + c = 2
Matrix = ⎢
⎥
⎣c d ⎦ c + d = 8 b + d = s
The four equations are solvable only if s=
Sol.
s=10
.