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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, threedimensional planes n v D 0, twodimensional
subspaces .n1 v D 0 and n2 v D 0/, onedimensional 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
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is not singular: not a subspace.
(b) The sum of singular matrices
C
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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
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CD
and A D
have the same column space. B D
has a
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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/.
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(or other examples).
(d) False: C .A I / ¤ C .A/ when A D I or A D
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#
#
"
#
"
"
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A D 1 0 0 and 1 0 1 do not have .1; 1; 1/ in C .A/. A D 2 4 0
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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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 Spring '12
 Minki
 Mass

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