Introduction to Linear algebra-Strang-Solutions-Manual_ver13

# The line 1 l of solutions contains v d 1 1 0 and w d 1

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Unformatted text preview: mn picture) are changed. If z D 2 then x C y D 0 and x y D z give the point .1; 1; 2/. If z D 0 then x C y D 6 and x y D 4 produce .5; 1; 0/. Halfway between those is .3; 0; 1/. If x; y; z satisfy the ﬁrst two equations they also satisfy the third equation. The line 1 L of solutions contains v D .1; 1; 0/ and w D . 1 ; 1; 1 / and u D 2 v C 1 w and all 2 2 2 combinations c v C d w with c C d D 1. Equation 1 C equation 2 equation 3 is now 0 D 4. Line misses plane; no solution. Column 3 D Column 1 makes the matrix singular. Solutions .x; y; z/ D .1; 1; 0/ or .0; 1; 1/ and you can add any multiple of . 1; 0; 1/; b D .4; 6; c/ needs c D 10 for solvability (then b lies in the plane of the columns). Four planes in 4-dimensional space normally meet at a point. The solution to Ax D .3; 3; 3; 2/ is x D .0; 0; 1; 2/ if A has columns .1; 0; 0; 0/; .1; 1; 0; 0/; .1; 1; 1; 0/, .1; 1; 1; 1/. The equations are x C y C z C t D 3; y C z C t D 3; z C t D 3; t D 2. (a) Ax D .18; 5; 0/ and (b) Ax D .3; 4; 5; 5/. Solutions to Exercises 8 10 Multiplying as linear combinations of the columns gives the same Ax . By rows or by 11 12 13 14 15 16 17 18 19 20 21 22 columns: 9 separate multiplications for 3 by 3. Ax equals .14; 22/ and .0; 0/ and (9; 7/. Ax equals .z; y; x/ and .0; 0; 0/ and (3; 3; 6/. (a) x has n components and Ax has m components (b) Planes from each equation in Ax D b are in n-dimensional space, but the columns are in m-dimensional space. 2x C 3y C z C 5t D 8 is Ax D b with the 1 by 4 matrix A D Œ 2 3 1 5 . The solutions x ﬁll a 3D “plane” in 4 dimensions. It could be called a hyperplane.     10 01 (a) I D (b) P D 01 10     01 1 0 ı ı 2 90 rotation from R D , 180 rotation from R D D I. 10 0 1 " # " # 010 001 P D 0 0 1 produces .y; z; x/ and Q D 1 0 0 recovers .x; y; z/. Q is the 100 010 inverse of P . " #   100 10 1 1 0 subtract the ﬁrst component from the second. ED and E D 11 001 " # # " 100 100 0 1 0 , E v D .3; 4; 8/ and E 1 E v recovers E D 0 1 0 and E 1 D 101 101 .3; 4; 5/.     00 10 projects onto the y -axis. projects onto the x -axis and P2 D P1 D 01 00    0 5 5 . and P2 P1 v D has P1 v D vD 0 0 7 p p 1 p2 p2 rotates all vectors by 45ı . The columns of R are the results from RD 2 2 2 rotating .1; 0/ and .0; 1/! "# x The dot product Ax D Œ 1 4 5  y D .1 by 3/.3 by 1/ is zero for points .x; y; z/ z on a plane in three dimensions. The columns of A are one-dimensional vectors. 23 A D Œ 1 2 I 3 4  and x D Œ 5 2  0 and b D Œ 1 7  0 . r D b A  x prints as zero. 24 A  v D Œ 3 4 5  0 and v 0  v D 50. But v  A gives an error message from 3 by 1 times 3 by 3. 25 ones.4; 4/  ones.4; 1/ D Œ 4 4 4 4  0 ; B  w D Œ 10 10 10 10  0 . 26 The row picture has two lines meeting at the solution (4; 2). The column picture will have 4.1; 1/ C 2. 2 ; 1/ D 4(column 1) C 2(column 2) D right side .0; 6/. 27 The row picture shows 2 planes in 3-dimensional space. The column picture is in 2-dimensional space. The solut...
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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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