Summary
Summary Introduction to Linear Algebra 5th Edition Solution Manual PDF
- Course
- Institution
- Book
PDF Solutions Manual for Introduction to Linear Algebra 5th Edition by Gilbert Strang.
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
Linear Algebra Practice Problems Set 5
-
Linear Algebra Practice Problems Set 4
-
Linear Algebra Practice Problems Set 1
Seller
Follow
SolutionsWizard
Reviews received
Content preview
, INTRODUCTIONTO
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
,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 ( 21 , 12 , 12 ).
Centers of faces are ( 12 , 21 , 0), ( 12 , 21 , 1) and (0, 12 , 12 ), (1, 21 , 12 ) and ( 12 , 0, 12 ), ( 12 , 1, 12 ).
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 π6 , sin π6 ) = ( 3/2, 1/2).
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.
1
17 All vectors cv + cw are on the line passing through (0, 0) and u = 2v + 12 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) 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.
1
20 (a) 3u + 13 v + 31 w is the center of the triangle between u, v and w; 21 u + 12 w lies
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 12 (u + v + w) is outside 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 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.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller SolutionsWizard. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
62890 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now