INTRODUCTION
TO
LINEAR
ALGEBRA
Fifth Edition
MANUAL FOR INSTRUCTORS
Gilbert Strang
Massachusetts Institute of Technology
,2 Solutions to Exercises
Problem Set 1.1, page 8
1 The combinations give (a) a line in R3 (b) a plane in R3 (c) all of R3.
2 v + w = (2, 3) and v − w = (6, −1) will be the diagonals of the parallelogram with
v and w as two sides going out from (0, 0).
3 This problem gives the diagonals v + w and v − w of the parallelogram and asks for
the sides: The opposite of Problem 2. In this example v = (3, 3) and w = (2, −2).
4 3v + w = (7, 5) and cv + dw = (2c + d, c + 2d).
5 u+v = (−2, 3, 1) and u+v+w = (0, 0, 0) and 2u+2v+w = ( add first answers) =
(−2, 3, 1). The vectors u, v, w are in the same plane because a combination gives (0,
0, 0). Stated another way: u = −v − w is in the plane of v and w.
6 The components of every cv + dw add to zero because the components of v and of w
add to zero. c = 3 and d = 9 give (3, 3, −6). There is no solution to cv+dw = (3, 3, 6)
because 3 + 3 + 6 is not zero.
7 The nine combinations c(2, 1) + d(0, 1) with c = 0, 1, 2 and d = (0, 1, 2) will lie on a
lattice. If we took all whole numbers c and d, the lattice would lie over the whole plane.
8 The other diagonal is v − w (or else w − v). Adding diagonals gives 2v (or 2w).
9 The fourth corner can be (4, 4) or (4, 0) or (−2, 2). Three possible parallelograms!
10 i − j = (1, 1, 0) is in the base (x-y plane). i + j + k = (1, 1, 1) is the opposite corner
from (0, 0, 0). Points in the cube have 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.
11 Four more corners (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1). The center point is ( 1 , 1 , 1 ).
2 2 2
Centers of faces are ( 1 , 1 , 0), ( 1 , 1 , 1) and (0, 1 , 1 ), (1, 1 , 1 ) and ( 1 , 0, 1 ), ( 1 , 1, 1 ).
2 2 2 2 2 2 2 2 2 2 2 2
12 The combinations of i = (1, 0, 0) and i + j = (1, 1, 0) fill the xy plane in xyz space.
13 Sum = zero vector. Sum = −2:00 vector = 8:00 vector. 2:00 is 30◦ from horizontal
√
= (cos π , sin π ) = ( 3/2, 1/2).
6 6
14 Moving the origin to 6:00 adds j = (0, 1) to every vector. So the sum of twelve vectors
changes from 0 to 12j = (0, 12).
,Solutions to Exercises 3
3 1
15 The point v+w is three-fourths of the way to v starting from w. The vector
4 4
1 1 1 1
v + w is halfway to u = v + w. The vector v + w is 2u (the far corner of the
4 4 2 2
parallelogram).
16 All combinations with c + d = 1 are on the line that passes through v and w.
The point V = −v + 2w is on that line but it is beyond w.
17 All vectors cv + cw are on the line passing through (0, 0) and u = 1 v + 1 w. That
2 2
line continues out beyond v + w and back beyond (0, 0). With c ≥ 0, half of this line
is removed, leaving a ray that starts at (0, 0).
18 The combinations cv + dw with 0 ≤ c ≤ 1 and 0 ≤ d ≤ 1 fill the parallelogram with
sides v and w. For example, if v = (1, 0) and w = (0, 1) then cv + dw fills the unit
square. But when v = (a, 0) and w = (b, 0) these combinations only fill a segment of
a line.
19 With c ≥ 0 and d ≥ 0 we get the infinite “cone” or “wedge” between v and w. For
example, if v = (1, 0) and w = (0, 1), then the cone is the whole quadrant x ≥ 0, y ≥
0. Question: What if w = −v? The cone opens to a half-space. But the combinations
of v = (1, 0) and w = (−1, 0) only fill a line.
20 (a) 1 u + 1 v + 1 w is the center of the triangle between u, v and w; 1 u + 1 w lies
3 3 3 2 2
between u and w (b) To fill the triangle keep c ≥ 0, d ≥ 0, e ≥ 0, and c + d + e = 1.
21 The sum is (v − u) +(w − v) +(u − w) = zero vector. Those three sides of a triangle
are in the same plane!
22 The vector 1 (u + v + w) is outside the pyramid because c + d + e = 1
+ 1 + 1 > 1.
2 2 2 2
23 All vectors are combinations of u, v, w as drawn (not in the same plane). Start by
seeing that cu + dv fills a plane, then adding ew fills all of R3.
24 The combinations of u and v fill one plane. The combinations of v and w fill another
plane. Those planes meet in a line: only the vectors cv are in both planes.
25 (a) For a line, choose u = v = w = any nonzero vector (b) For a plane, choose
u and v in different directions. A combination like w = u + v is in the same plane.
, 4 Solutions to Exercises
26 Two equations come from the two components: c + 3d = 14 and 2c + d = 8. The
solution is c = 2 and d = 4. Then 2(1, 2) + 4(3, 1) = (14, 8).
27 A four-dimensional cube has 24 = 16 corners and 2 · 4 = 8 three-dimensional faces
and 24 two-dimensional faces and 32 edges in Worked Example 2.4 A.
28 There are 6 unknown numbers v1, v2, v3, w1, w2, w3. The six equations come from the
components of v + w = (4, 5, 6) and v − w = (2, 5, 8). Add to find 2v = (6, 10, 14)
so v = (3, 5, 7) and w = (1, 0, −1).
29 Two combinations out of infinitely many that produce b = (0, 1) are −2u + v and
1 1
2
w − 2
v. No, three vectors u, v, w in the x-y plane could fail to produce b if all
three lie on a line that does not contain b. Yes, if one combination produces b then two
(and infinitely many) combinations will produce b. This is true even if u = 0; the
combinations can have different cu.
30 The combinations of v and w fill the plane unless v and w lie on the same line through (0,
0). Four vectors whose combinations fill 4-dimensional space: one example is the
“standard basis” (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1).
31 The equations cu + dv + ew = b are
2c −d =1 So d = 2e c = 3/4
−c +2d −e = 0 then c = 3e d = 2/4
−d +2e = 0 then 4e = 1 e = 1/4
Problem Set 1.2, page 18
1 u · v = −2.4 + 2.4 = 0, u · w = −.6 + 1.6 = 1, u · (v + w) = u · v + u · w =
0 + 1, w · v = 4 − 6 = −2 = v · w.
√
2 ǁuǁ = 1 and ǁvǁ = 5 and ǁwǁ = 5. Then |u · v| = 0 < (1)(5) and |v · w| = 10 <
√
5 5, confirming the Schwarz inequality.
TO
LINEAR
ALGEBRA
Fifth Edition
MANUAL FOR INSTRUCTORS
Gilbert Strang
Massachusetts Institute of Technology
,2 Solutions to Exercises
Problem Set 1.1, page 8
1 The combinations give (a) a line in R3 (b) a plane in R3 (c) all of R3.
2 v + w = (2, 3) and v − w = (6, −1) will be the diagonals of the parallelogram with
v and w as two sides going out from (0, 0).
3 This problem gives the diagonals v + w and v − w of the parallelogram and asks for
the sides: The opposite of Problem 2. In this example v = (3, 3) and w = (2, −2).
4 3v + w = (7, 5) and cv + dw = (2c + d, c + 2d).
5 u+v = (−2, 3, 1) and u+v+w = (0, 0, 0) and 2u+2v+w = ( add first answers) =
(−2, 3, 1). The vectors u, v, w are in the same plane because a combination gives (0,
0, 0). Stated another way: u = −v − w is in the plane of v and w.
6 The components of every cv + dw add to zero because the components of v and of w
add to zero. c = 3 and d = 9 give (3, 3, −6). There is no solution to cv+dw = (3, 3, 6)
because 3 + 3 + 6 is not zero.
7 The nine combinations c(2, 1) + d(0, 1) with c = 0, 1, 2 and d = (0, 1, 2) will lie on a
lattice. If we took all whole numbers c and d, the lattice would lie over the whole plane.
8 The other diagonal is v − w (or else w − v). Adding diagonals gives 2v (or 2w).
9 The fourth corner can be (4, 4) or (4, 0) or (−2, 2). Three possible parallelograms!
10 i − j = (1, 1, 0) is in the base (x-y plane). i + j + k = (1, 1, 1) is the opposite corner
from (0, 0, 0). Points in the cube have 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.
11 Four more corners (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1). The center point is ( 1 , 1 , 1 ).
2 2 2
Centers of faces are ( 1 , 1 , 0), ( 1 , 1 , 1) and (0, 1 , 1 ), (1, 1 , 1 ) and ( 1 , 0, 1 ), ( 1 , 1, 1 ).
2 2 2 2 2 2 2 2 2 2 2 2
12 The combinations of i = (1, 0, 0) and i + j = (1, 1, 0) fill the xy plane in xyz space.
13 Sum = zero vector. Sum = −2:00 vector = 8:00 vector. 2:00 is 30◦ from horizontal
√
= (cos π , sin π ) = ( 3/2, 1/2).
6 6
14 Moving the origin to 6:00 adds j = (0, 1) to every vector. So the sum of twelve vectors
changes from 0 to 12j = (0, 12).
,Solutions to Exercises 3
3 1
15 The point v+w is three-fourths of the way to v starting from w. The vector
4 4
1 1 1 1
v + w is halfway to u = v + w. The vector v + w is 2u (the far corner of the
4 4 2 2
parallelogram).
16 All combinations with c + d = 1 are on the line that passes through v and w.
The point V = −v + 2w is on that line but it is beyond w.
17 All vectors cv + cw are on the line passing through (0, 0) and u = 1 v + 1 w. That
2 2
line continues out beyond v + w and back beyond (0, 0). With c ≥ 0, half of this line
is removed, leaving a ray that starts at (0, 0).
18 The combinations cv + dw with 0 ≤ c ≤ 1 and 0 ≤ d ≤ 1 fill the parallelogram with
sides v and w. For example, if v = (1, 0) and w = (0, 1) then cv + dw fills the unit
square. But when v = (a, 0) and w = (b, 0) these combinations only fill a segment of
a line.
19 With c ≥ 0 and d ≥ 0 we get the infinite “cone” or “wedge” between v and w. For
example, if v = (1, 0) and w = (0, 1), then the cone is the whole quadrant x ≥ 0, y ≥
0. Question: What if w = −v? The cone opens to a half-space. But the combinations
of v = (1, 0) and w = (−1, 0) only fill a line.
20 (a) 1 u + 1 v + 1 w is the center of the triangle between u, v and w; 1 u + 1 w lies
3 3 3 2 2
between u and w (b) To fill the triangle keep c ≥ 0, d ≥ 0, e ≥ 0, and c + d + e = 1.
21 The sum is (v − u) +(w − v) +(u − w) = zero vector. Those three sides of a triangle
are in the same plane!
22 The vector 1 (u + v + w) is outside the pyramid because c + d + e = 1
+ 1 + 1 > 1.
2 2 2 2
23 All vectors are combinations of u, v, w as drawn (not in the same plane). Start by
seeing that cu + dv fills a plane, then adding ew fills all of R3.
24 The combinations of u and v fill one plane. The combinations of v and w fill another
plane. Those planes meet in a line: only the vectors cv are in both planes.
25 (a) For a line, choose u = v = w = any nonzero vector (b) For a plane, choose
u and v in different directions. A combination like w = u + v is in the same plane.
, 4 Solutions to Exercises
26 Two equations come from the two components: c + 3d = 14 and 2c + d = 8. The
solution is c = 2 and d = 4. Then 2(1, 2) + 4(3, 1) = (14, 8).
27 A four-dimensional cube has 24 = 16 corners and 2 · 4 = 8 three-dimensional faces
and 24 two-dimensional faces and 32 edges in Worked Example 2.4 A.
28 There are 6 unknown numbers v1, v2, v3, w1, w2, w3. The six equations come from the
components of v + w = (4, 5, 6) and v − w = (2, 5, 8). Add to find 2v = (6, 10, 14)
so v = (3, 5, 7) and w = (1, 0, −1).
29 Two combinations out of infinitely many that produce b = (0, 1) are −2u + v and
1 1
2
w − 2
v. No, three vectors u, v, w in the x-y plane could fail to produce b if all
three lie on a line that does not contain b. Yes, if one combination produces b then two
(and infinitely many) combinations will produce b. This is true even if u = 0; the
combinations can have different cu.
30 The combinations of v and w fill the plane unless v and w lie on the same line through (0,
0). Four vectors whose combinations fill 4-dimensional space: one example is the
“standard basis” (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1).
31 The equations cu + dv + ew = b are
2c −d =1 So d = 2e c = 3/4
−c +2d −e = 0 then c = 3e d = 2/4
−d +2e = 0 then 4e = 1 e = 1/4
Problem Set 1.2, page 18
1 u · v = −2.4 + 2.4 = 0, u · w = −.6 + 1.6 = 1, u · (v + w) = u · v + u · w =
0 + 1, w · v = 4 − 6 = −2 = v · w.
√
2 ǁuǁ = 1 and ǁvǁ = 5 and ǁwǁ = 5. Then |u · v| = 0 < (1)(5) and |v · w| = 10 <
√
5 5, confirming the Schwarz inequality.
