{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Introduction to Linear algebra-Strang-Solutions-Manual_ver13

# Introduction to Linear...

This preview shows pages 1–3. Sign up to view the full content.

INTRODUCTION TO LINEAR ALGEBRA Fourth 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: [email protected] Wellesley - Cambridge Press Box 812060 Wellesley, Massachusetts 02482

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Solutions to Exercises Problem Set 1.1, page 8 1 The combinations give (a) a line in R 3 (b) a plane in R 3 (c) all of R 3 . 2 v C w D .2; 3/ and v NUL w D .6; NUL 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 C w and v NUL w of the parallelogram and asks for the sides: The opposite of Problem 2. In this example v D .3; 3/ and w D .2; NUL 2/ . 4 3 v C w D .7; 5/ and c v C d w D .2c C d; c C 2d/ . 5 u C v D . NUL 2; 3; 1/ and u C v C w D .0; 0; 0/ and 2 u C 2 v C w D . add first answers / D . NUL 2; 3; 1/ . The vectors u ; v ; w are in the same plane because a combination gives .0; 0; 0/ . Stated another way: u D NUL v NUL w is in the plane of v and w . 6 The components of every c v C d w add to zero. c D 3 and d D 9 give .3; 3; NUL 6/ . 7 The nine combinations c.2; 1/ C d.0; 1/ with c D 0; 1; 2 and d 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 NUL w (or else w NUL v ). Adding diagonals gives 2 v (or 2 w ). 9 The fourth corner can be .4; 4/ or .4; 0/ or . NUL 2; 2/ . Three possible parallelograms! 10 i NUL j D .1; 1; 0/ is in the base ( x - y plane). i C j C k D .1; 1; 1/ is the opposite corner from .0; 0; 0/ . Points in the cube have 0 ² x ² 1 , 0 ² y ² 1 , 0 ² z ² 1 . 11 Four more corners .1; 1; 0/; .1; 0; 1/; .0; 1; 1/; .1; 1; 1/ . The center point is . 1 2 ; 1 2 ; 1 2 / . Centers of faces are . 1 2 ; 1 2 ; 0/; . 1 2 ; 1 2 ; 1/ and .0; 1 2 ; 1 2 /; .1; 1 2 ; 1 2 / and . 1 2 ; 0; 1 2 /; . 1 2 ; 1; 1 2 / . 12 A four-dimensional cube has 2 4 D 16 corners and 2 ± 4 D 8 three-dimensional faces and 24 two-dimensional faces and 32 edges in Worked Example 2.4 A . 13 Sum D zero vector. Sum D NUL 2 : 00 vector D 8 : 00 vector. 2 : 00 is 30 ı from horizontal D . cos ± 6 ; sin ± 6 / D . p 3=2; 1=2/ . 14 Moving the origin to 6 : 00 adds j D .0; 1/ to every vector. So the sum of twelve vectors changes from 0 to 12 j D .0; 12/ . 15 The point 3 4 v C 1 4 w is three-fourths of the way to v starting from w . The vector 1 4 v C 1 4 w is halfway to u D 1 2 v C 1 2 w . The vector v C w is 2 u (the far corner of the parallelogram). 16 All combinations with c C d D 1 are on the line that passes through v and w . The point V D NUL v C 2 w is on that line but it is beyond w . 17 All vectors c v C c w are on the line passing through .0; 0/ and u D 1 2 v C 1 2 w . That line continues out beyond v C w and back beyond .0; 0/ . With c ³ 0 , half of this line is removed, leaving a ray that starts at .0; 0/ . 18 The combinations c v C d w with 0 ² c ² 1 and 0 ² d ² 1 fill the parallelogram with sides v and w . For example, if v D .1; 0/ and w D .0; 1/ then c v C d w fills the unit square.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 99

Introduction to Linear...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online