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INTRODUCTION

TO

LINEAR

ALGEBRA

Fifth Edition

MANUAL FOR INSTRUCTORS

Gilbert Strang

Massachusetts Institute of Technology

math.mit.edu/linearalgebra

web.mit.edu/18.06

video lectures: ocw.mit.edu

math.mit.edu/∼gs

www.wellesleycambridge.com

email: linearalgebrabook@gmail.com

Wellesley-Cambridge Press

Box 812060

Wellesley, Massachusetts 02482

Solutions to Exercises

3

15 The point

3

4

v +

1

4

w is three-fourths of t he way to v starting from w. The vector

1

4

v +

1

4

w is halfway to u =

1

2

v +

1

2

w. The vector v + w is 2u (the far corner of the

parallelogram).

16 All combinations with c + d = 1 are o n the line that passes through v and w.

The p oint V = − v + 2w is o n th at line but it is beyond w.

17 All vectors cv + cw are on the line passing t hrough (0, 0) and u =

1

2

v +

1

2

w. That

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) t hese combinat ions only ﬁll a segment of

a l ine.

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 q uadrant x ≥ 0, y ≥

0. Question: What if w = −v? The cone o pens t o a half-space. But the combinations

of v = (1, 0) and w = (−1, 0) only ﬁll a l ine.

20 (a)

1

3

u +

1

3

v +

1

3

w is the center of the triang le between u, v and w;

1

2

u +

1

2

w li es

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

2

(u + v + w) is o utside the pyramid because c + d + e =

1

2

+

1

2

+

1

2

> 1.

23 All vectors are combinations of u, v, w as drawn (not in the same plane). Start by

seeing that cu + dv ﬁ lls a plan e, then adding ew ﬁlls all of R

3

.

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 directio ns. A combination like w = u + v is in the same plane.

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