Introduction to Linear algebra-Strang-Solutions-Manual_ver13

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

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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 &gt; 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 uv 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 &lt; .1/.5/ and 48 &lt; .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....
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## 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.

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