Introduction to Linear algebra-Strang-Solutions-Manual_ver13

A solution for every b b solvable only if b3 d 0 c

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Unformatted text preview: R2 itself, lines through .0; 0/, and .0; 0/ by itself (b) The subspaces of R4 are R4 itself, three-dimensional planes n  v D 0, two-dimensional subspaces .n1  v D 0 and n2  v D 0/, one-dimensional lines through .0; 0; 0; 0/, and .0; 0; 0; 0/ by itself. (a) Two planes through .0; 0; 0/ probably intersect in a line through .0; 0; 0/ (b) The plane and line probably intersect in the point .0; 0; 0/ (c) If x and y are in both S and T , x C y and c x are in both subspaces. The smallest subspace containing a plane P and a line L is either P (when the line L is in the plane P) or R3 (when L is not in P). (a) The invertible matrices do not include the zero matrix, so they are not a subspace    10 00 is not singular: not a subspace. (b) The sum of singular matrices C 00 01 Solutions to Exercises 28 18 (a) True: The symmetric matrices do form a subspace 19 20 21 22 23 (b) True: The matrices with AT D A do form a subspace (c) False: The sum of two unsymmetric matrices could be symmetric. The column space of A is the x -axis D all vectors .x; 0; 0/. The column space of B is the xy plane D all vectors .x; y; 0/. The column space of C is the line of vectors .x; 2x; 0/. (a) Elimination leads to 0 D b2 2 b1 and 0 D b1 C b3 in equations 2 and 3: Solution only if b2 D 2b1 and b3 D b1 (b) Elimination leads to 0 D b1 C 2b3 in equation 3: Solution only if b3 D b1 . A combination of the columns of C is also a combination of the columns of A. Then       13 12 12 CD and A D have the same column space. B D has a 26 24 36 different column space. (a) Solution for every b (b) Solvable only if b3 D 0 (c) Solvable only if b3 D b2 . The extra column b enlarges the column space unless b is already in the column space.     1 0 1 (larger column space) 1 0 1 (b is in column space) ŒA b D 0 0 1 (no solution to Ax D b) 0 1 1 (Ax D b has a solution) 24 The column space of AB is contained in (possibly equal to) the column space of A. 25 26 27 28 29 30 31 32 The example B D 0 and A ¤ 0 is a case when AB D 0 has a smaller column space than A. The solution to Az D b C b is z D x C y . If b and b are in C .A/ so is b C b . The column space of any invertible 5 by 5 matrix is R5 . The equation Ax D b is always solvable (by x D A 1 b/ so every b is in the column space of that invertible matrix. (a) False: Vectors that are not in a column space don’t form a subspace. (b) True: Only the zero matrix has C .A/ D f0g. (c) True: C .A/ D C .2A/.  10 (or other examples). (d) False: C .A I / ¤ C .A/ when A D I or A D 00 # # " # " " 120 112 110 A D 1 0 0 and 1 0 1 do not have .1; 1; 1/ in C .A/. A D 2 4 0 360 010 011 has C .A/ D line. When Ax D b is solvable for all b, every b is in the column space of A. So that space is R9 . (a) If u and v are both in S C T , then u D s1 C t 1 and v D s2 C t 2 . So u C v D .s1 C s2 / C .t 1 C t 2 / is also in S C T . And so is c u D c s1 C c t 1 : a subspace. (b) If S and T are different lines, then S [ T is just the two lines (not a subspace) but S C T is the whole plane that they span. If S D C .A/ and T D C .B/ then S C T is the column space of M D Œ A B . The columns of AB are combinations  the columns of...
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