Unformatted text preview: MAT223 Final
December 2004
Note: I’ll often write column vectors horziontally and with commas to save space.
Part I
1. (2). Solving for X , X = (AT)− 1A − 1 = (A AT) −1. Now, det A = 1, so det(A AT) = 1, which
ab
d −b
means that the inverse of A AT =
is just
, and we only need to compute
cd
−c a
1
a=[1 −1]
= 2.
−1
2. (4). In A we have the solutions of a homogeneous system of linear equations, so there we
do have a subspace. In B the system is not linear and though in C it is, it’s not homogeneous. 0 −1 1
3. (1). In the basis {1, x, x2}, the vectors in S are the rows of − 1 1 0 whose determi1 k1
nant is − (k + 2). Thus S is a basis unless k = − 2. (Alternatively, use rowreduction.)
4. (1). The given basis for W is orthogonal, so we can compute X · V2
1 1 1 1 1 X · V1
V1 +
V2 = − 1 + 1 = 0 .
projW X =
V2 2
20
V1 2
20
0
5. (5). X = V3 − V3 · V1
V1 2 V1 + V3 · V2
V2 2 1 2 1 V2 = [1, 1, 0, 1] − 3 [1, − 1, 0, 1] − 2 [1, 1, 0, 0] = 3 [ − 1, 1, 0, 2]. 6. (2). You get the elemntary matrix by applying the same row operation to the identity
matrix.
7. (6). By Cramer’s Rule, x2 = a − 2b c
d − 2e f
g − 2h k / abc
def
ghk =−2 abc
def
ghk / abc
def
ghk = − 2. 8. (5). A is false since the eigenvalues of A− 1 are λ− 1 where λ is an eigenvalue of A, so they
are usually diﬀerent from those of A. B is true: if A v = λ v , then A− 1v = λ− 1v , so (A− 1 +
1+λ
I )v = λ− 1v + v = λ v . C is false, not every invertible matrix is diagonalizable.
9. (5). Expanding by the second row, det A = − 2k
2 2 − 2k (k − 1) (k + 1) − 1 = 2k (k − 1)(k + 2). 1
k−1 k+1
k + 1 k2
1
0
k−1
0 = 2k (k − 1) 1
k+1
k+1
1 = 10. (4). A is true: if A B is invertible, B − 1 = (A B )− 1A exists. B is also true: if A B = −
(A B )T, taking determinants on both sides, det A det B = ( − 1)3 det A det B , and since
det A 0, it follows that det B = 0. C is false: if A B = B A, B could be the zero matrix,
for example.
11. (6). Since dim(im T ) + dim(ker T ) = dim(domain) = 7, option A is correct. B is not
because, the rank of A is dim(im T ), not 8 minus that. Finally, C is also correct because
rank A = dim(row A) = 7 − dim((row A)⊤).
12. (1). The third column of the standard matrix representation is just T ([0, 0, 1]) = 2[0, 0,
1] − N = [ − 1, 0, 1].
1 Part II
13.
a) If B is not invertible, det B = 0, that is, a d − b c = 0. Therefore,
B2 = a2 + a d a b + b d
a2 + b c a b + b d
ab
= (a + d)
= (a + d)B.
=
2
cd
a c + c d a d + d2
ac+cd bc+d b) Assume that t1 X1 + t2 X2 + t3 X3 = 0, we must prove t1 = t2 = t3 = 0. Well, multiplying by the matrix A we get t1 A X1 + t2 A X2 + t3 A X3 = 0. Since {A X1, A X2,
A X3} is assumed to be independent, we get t1 = t2 = t3 = 0 as desired.
The converse is false in general, for example if A is the zero matrix, clearly {A X1,
A X2, A X3} will de linearly dependent no matter what.
14.
a) Note that W ⊥ is given as a solution set of a system of homogeneous linear equations, thus we can write it as the null space of the matrix of coeﬃcients of said 1 0 −1 0 0 1 1 0 0 −1 system: W ⊥ = null A where A = 0 1 1 0 − 1 . So we want a basis for W =
0 0 0 1 −1
⊥⊥
⊥
(W ) = (null A) . I claim that (null A)⊥ = row A. Consider an arbitrary X ∈
null A. To calculate the product A X , we take the dot product of each row of A
with the column vector X , and since X ∈ null A, all of these products must be
zero. This means that each row of A belongs to (null A)⊥, and so, row A ⊆
(null A)⊥. To prove those subspaces are equal it is enough to show they have the
same dimension, but dim (null A)⊥ = 5 − dim null A = rank A = dim row A.
So, we have W = row A. To ﬁnd a basis for W , we simply use row reduction: 1
1 0
0 0 −1
10
11
00 0
0
0
1 0
1 −1 R2 − R1 0
0
−1 −1
0 0 −1
11
11
00 0
0
0
1 0
1 −1 R3 − R2 0
0
−1 −1
0 0 −1
11
00
00 0
0
0
1 0
−1 ,
0
−1 so we see that {[1, 0, − 1, 0, 0], [0, 1, 1, 0, − 1], [0, 0, 0, 1, − 1]} is a basis for W .
b) Let X ∈ W ⊤, we must show X ∈ U ⊤. For this we have to show that X · Y = 0 for
any Y ∈ U . But any Y ∈ U is also in W (since U ⊆ W ), and since X ∈ W ⊤, we do
have X · Y = 0.
15.
a) We need eigenvectors corresponding to the eigenvalues 2 and − 1. First, let’s do
the eigenvalue − 1: we need to solve (A − ( − 1)I )X = 0, that is, (A + I )X = 0. The
three equations in the system are the same, namely, x + y + z = 0. The general
solution, of course, is given by x = t1, y = t2, z = − t1 − t2. We ﬁnd a basis of the
solution space by setting t1 = 1, t2 = 0 to get [1, 0, − 1] and then t1 = 0, t2 = 1 to get
[0, 1, − 1].
No for the eigenvalue 2: we solve the system (A − 2I )X = 0 by row reduction: 1 −2 1
1 − 2 1 R3 + R2 1 − 2 1
−2 1 1 1 − 2 1 R1 ↔ R2 − 2 1 1 R2 + 2R1 0 − 3 3 1 0 1 − 1 R3 − R1
1 1 −2
0 3 − 3 − 3 R2 0 0 0
1 1 −2 2 which means that the solution space is given by z = t, y = t, x = t, so [1, 1, 1] is a
basis for it. 1 01
Therefore, writing our eigenvectors as columns, P = 0 1 1 , and we’ll have
−1 −1 1
−1 0 0
P − 1A P = 0 − 1 0 . A is diagonalizable because we found a basis of eigenvec0 02
tors: the columns of P . a00
b) Let P be such that P − 1A P is a diagonal matrix, say 0 b 0 . Then we have
00c
3 a3 0 0
a00 0 b3 0 = 0 b 0 = (P − 1A P )3 = P − 1A3P = 0,
00c
0 0 c3
so a3 = b3 = c3 = 0. But this implies that a = b = c = 0, and so P − 1A P = 0. Solving
for A, A = P 0 P − 1 = 0.
16.
a) Clearly both A and B have two dimensional columns spaces. By either row reduction, or a little trial and error, we ﬁnd that the columns of B are linear combinations of the columns of A: B1 = 2A1 − A2 and B2 = A1 + A2. This tells us that
col B ⊆ col A. Since both spaces have dimension two, they must be equal.
b) Let X ∈ null A ∩ col A. Since X ∈ col A = im A, we know that X = A Y for some
vector Y . We also have X ∈ null A, so that A X = 0, i.e., A2Y = 0, so that Y ∈
null A2 = null A. Therefore, X = A Y = 0, as desired. 3 ...
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 Spring '09
 UPPAL
 Linear Algebra, Algebra, Vectors, basis, Row, nant

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