Introduction to Linear algebra-Strang-Solutions-Manual_ver13

For example if v d 1 0 and w d 0 1 then the cone is

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: or “wedge” between v and w. For example, if v D .1; 0/ and w D .0; 1/, then the cone is the whole quadrant x  0, y  0. Question: What if w D v? The cone opens to a half-space. 13 Sum D zero vector. Sum D p   14 15 16 17 18 19 Solutions to Exercises 3 1 C 1 v C 1 w is the center of the triangle between u; v and w; 1 u C 2 w lies 3 3 2 between u and w (b) To fill the triangle keep c  0, d  0, e  0, and c C d C e D 1. 20 (a) 1 u 3 21 The sum is .v u / C .w are in the same plane! v / C .u w/ D zero vector. Those three sides of a triangle 1 22 The vector 2 .u C v C w/ is outside the pyramid because c C d C e D 1 2 C 1 C 1 > 1. 2 2 23 All vectors are combinations of u; v; w as drawn (not in the same plane). Start by seeing that c u C d v fills a plane, then adding e w 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 c v are in both planes. 25 (a) For a line, choose u D v D w D any nonzero vector (b) For a plane, choose u and v in different directions. A combination like w D u C v is in the same plane. 26 Two equations come from the two components: c C 3d D 14 and 2c C d D 8. The solution is c D 2 and d D 4. Then 2.1; 2/ C 4.3; 1/ D .14; 8/. 27 The combinations of i D .1; 0; 0/ and i C j D .1; 1; 0/ fill the xy plane in xyz space. 28 There are 6 unknown numbers v1 ; v2 ; v3 ; w1 ; w2 ; w3 . The six equations come from the components of v C w D .4; 5; 6/ and v so v D .3; 5; 7/ and w D .1; 0; 1/. w D .2; 5; 8/. Add to find 2v D .6; 10; 14/ 29 Two combinations out of infinitely many that produce b D .0; 1/ are 2u C v and 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 D 0; the combinations can have different c u. 1 w 2 1 v. 2 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 c u C d v C e w D b are 2c d D1 c C2d eD0 d C2e D 0 So d D 2e then c D 3e then 4e D 1 c D 3=4 d D 2=4 e D 1=4 Problem Set 1.2, page 19 1 uv D 1:8 C 3:2 D 1:4, u  w D 4 :8 C 4:8 D 0, v  w D 24 C 24 D 48 D w  v. 2 kuk D 1 and kvk D 5 and kwk D 10. Then 1:4 < .1/.5/ and 48 < .5/.10/, confirming the Schwarz inequality. 3 Unit vectors v=kvk D . 3 ; 4 / D .:6; :8/ and w=kwk D . 4 ; 3 / D .:8; :6/. The cosine 55 55 of  is kvk  kwk D 24 . The vectors w; u; w make 0ı ; 90ı ; 180ı angles with w. v w 25 4 (a) v  . v/ D 1 (b) .v C w/  .v w/ D v  v C w  v v  w w  w D 1 C . / . / 1 D 0 so  D 90ı (notice v  w D w  v) (c) .v 2w/  .v C 2w/ D v  v 4w  w D 1 4 D 3. Solutions to Exercises 4 p p 5 u1 D v=kvk D .3; 1/= 10 and u2 D p=kwk D .2; 1; 2/=3....
View Full Document

This note was uploaded on 09/25/2012 for the course PHY 103 taught by Professor Minki during the Spring '12 term at Korea Advanced Institute of Science and Technology.

Ask a homework question - tutors are online