matrizes, eliminação gaussiana, equações lineares e espaços vetoriais
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A ~ [~ ~] and B ~ [~ acompute AB and BA and A-I and B-1 and (AB)-I.
1.3 Find examples of 2 by 2 matrices with u 12 = ~ for which(a) A2=! (b) A-1=A1 (c) A2=A.
Solve by elimination and back-substitution:u +w=4 I'+w=o
+ W =0
1.5 Factor the preceding matrices into A = LV or PA = LV.
1.6 (a) There are sixteen 2 by 2 matrices whose entries are l's and O's. How many areinvertible?(b) (Much harder!) If you put 1's and O's at random into the entries of a 10 by 10matrix, is it more likely to be invertible or singular?
1.7 There are sixteen 2 by 2 matrlces whose entries are 1's and -1's. How many areinvertible?
E = r ~ ~ ~J or E = [1
1 1] or E = r ~ ~ 0oll?l4 0 I 00 Olio1.9 Write down a 2 by 2 system with infinitely many solutions.e Find inverses if they exist, by inspection or by Gauss-Jordan:
o 1J1 0
I IA = r ~ ~ ~ J and A = r ~ - ~ - ~J.lo 1 2 l-2 I I
1.11 If E is 2 by 2 and it adds the [lrsl equation to the second, what are £2 an £ . ~8E?8 True or false, with reason if true or counterexample if false:(1) If A lS invertible and its rows are in reverse order in B. then B .(2) If A and B are symmetric then AB it> symmetric.(3) If A and B are invertible then BA is invertible.
(4) Every nonsingular matrix can be factored into the product A = LV of a lowertriangular L and an upper triangular V.
1.13 Solve Ax = h by solving the triangular systems Lc = h and V x = c:
II ° OJ l2 2 41 lOJA = LV = 4 1 ° ° 1 3, h = 0 .
I ° I ° ° 1 IWhat part of A - I have you found, with this particular h?8 If possible, find 3 by 3 matrlces B such that
. (a) BA = 2A for every A(b) BA = 2B for every A(c) BA has the lirst and last rows of A reversed(d) BA has the first and last columns of A reversed.8Find the value for c in the following n by n inverse:
n -I -1 c-I n -1 I c
if A = then A -I = ---I n + 1
-I -1 -18 For which values of k does
kx + Y = 1x + ky = 1
[
1 2A = 2 6
o 4
I VI 0 00 Vz 0 °A=0 v3 1 00 V4 0 1
(a) Factor A into LV, assuming U2 =IeO.(b) Find A-I, which has the same form as A.8Solve by elimination, or show that there is no solution:
u+ V+ w=O u+ V+ w=Ou + 2v + 3w = 0
3u + 5u + 7w = 1
u + v + Jw = 0
Ju + 5v + 7w = I.
Find a basis for the following subspaces of R4:
(a) The vectors for which x I = 2x4
(b) The vectors for which XI + -\2 + .'(3 = a and XJ + -\4 = 0(c) The subspace spanned by (1,1. I, I), (I, 2, 3.4), and (2,3,4,5).
By giving a basis, describe a two-dimensional subspace of R 3 that contains none ofthe coordinate vectors (I, 0, 0), (0,1, 0), (0, 0, I).
True or false, with counterexample if false:
(i) [f the vectors XI' ... , Xm span a subspace 5, then dim 5 = m.(ii) The intersection of two subspaces of a vector space cannot be empty.(iii) [fAx = Ay, then x = y.(iv) The row space of A has a unique basis that can be computed by reducing A to
echelon form.(v) [f a square matrix A has independent columns, so does A 2.
G What is the echelon form U of
A ~r-: 2 a 2 ~}-2 1 12 -3 -7 -2
2.5 Find the rank and the nullspace of
A ~ r~a :l B = r~0
2l0 and 0~ .
1 1e Find bases for the four fundamental subspaces associated with
A = [~ a B = [~ ~l c = [~0 a0
What is the most general solution to u + r + w = 1, u - w = 2?
(a) Construct a matrix whose nullspace contains the vector x = (1, 1,2).(b) Construct a matrix whose left nullspace contains y = (1, 5).(c) Construct a matrix whose col umn space is spanned by (1, 1, 2) and whose rowspace is spanned by (l, 5).(d) [f you are given any three vectors in R6 and any three vectors in R5, is there a6 by 5 matrix whose column space is spanned by the first three and whose row spaceis spanned by the second three?
(n the vector space of 2 by 2 matrices,(a) is the set of rank-one matrices a subspace?(b) what subspace is spanned by the permutation matrices?(c) what subspace is spanned by the positive matrices (all au > a)?(d) what subspace is spanned by the invertible matrices?
2.10 Invent a vector space that contains all linear transformations from Rn to Rn. You have
to decide on a rule for addition. What is its dimension'l
I2 I
A = LV =12 1
3 2 4
12021002 200o 0 000 1000 0 0 0
(b) T F The first 3 rows of V are a basis for the row space of AT F Columns I, 3, 6 of V are a basis for the column space of AT F The four rows of A are a basis for the row space of A
(c) Find as many linearly independent vectors h as possible for which Ax = h has asolution.(dl In elimination on A, what multiple of the third row is subtracted to knock outthe fourth row?
2.12 If A is an 11 by 11 - 1 matrix, and its rank is 11 - 2, what is the dimension of itsnullspaee?
2.13 Use elimination to find the triangular factors in A = LV, if
a a a a
a h b hA=
a h c C
(! h C d
Under what conditions on the numbers a, h, c, d are the columns linearly independent?e Do the vectors (I, 1,3), (2, 3,6), and (1,4,3) form a basis for R"'?
2.15 Give examples of matrices A for which the number of solutions to Ax = h is
(i) 0 or I, depending on h;(ii) 'Xl, independent of h;
(iii) 0 or w, depending on h;(iv) I, regardless of h.
2.16 In the previous exercise, how is r related to m and 11 in each example?
2.17 If.x is a vector in R", and .xry = 0 for every y, prove that x = o.8If A is an n by IJ matrix such that A 2 = A and rank A = 11, prove that A = l.
2.19 What subspace of 3 by 3 matrices is spanned by the elementary matrices Eij, withones on the diagonal and at most one nonzero entry below?
2.20 How many 5 by 5 permutation matrices are there? Are they linearly independent'.Do they span the space of all 5 by 5 matrices? No need to write them all down.
2.21 What is the rank of the /1 by 11 matrix with every entry equal to one? How about he"checkerboard matrix," with aij = 0 when i +.i is even, ([I) = I when i +.i is odd?
@ (a) Ax = h has a solution under what conditions on h, if
A =[~ ~ ~ ~land h ~ r::Jl?2 4 a 1 LD)
(b) Find a basis for the nullspace of A.(c) Find the general solution to Ax = D, when a solution exists.(d) Find a basis for the column space of A.(e) What is the rank of AT'~
2.23 How can you construct a matrix which transforms the coordinate vectors el• e2• eJ
into three given vectors L'l, [2' v3? When will that matrix be invertible?8If e1• e2, e3 are in the column space of a 3 by 5 matrix, does it have a left-inverse?Does it have a right-inverse?
2.25 Suppose T is the linear transformation on R3 that takes each point (u, v, 1'\1) to(u + (;+ W, 14 + v, 14). Describe what T- 1 does to the point (x, y, z).
2.26 True orfal:;e: (a) Every subspace of R4 is the nullspace of some matrix.(b) If A has the same nullspace as AT, the matrix must be square.(c) The transformation that takes x to mx + h is linear (from R I to R 1).e Find bases for the four fundamental subspaces of
120 3a 2 2 2a a 0 aa a a 4
2.28 (a) If the rows of A are linearly independent (A is m by 11) then the rank is _and the column space is ._ _ and the left nullspace is _(b) If A is 8 by 10 with a 2-dimensional nullspace, show that Ax = h can be solvedfor every D.
2.29 Describe the linear transformations of the x-y plane that are represented with stan-dard basis (I, 0) and (0, 1) by the matrices
A3 = [ 0 1].-I 0
2.30 (a) If A is square, show that the nullspace of A2 contains the nullspace of A.(b) Show also that the column space of A 2 is contained in the column space of A.
2.32 (a)(b)(c)
Find a basis for the space of all vectors in RCJ with Xl + X2 = -'3 +.\4 = -'s;- .\0'
Find a matrix with that subspace as its nullspace.Find a matrix with that subspace as its column space.