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Page 146 Chapter 3 True False Questions.
1. The image of a 3x4 matrix is a subspace of
R 4?
False. It is a subspace of R3.
2. The span of vectors V1, V2, …,Vn consists of
all linear combinations of vectors V1, V2, …, Vn.
True. That is the definition of the span.
3.
If V1, V2, …, Vn are linearly independent
vectors in Rn, then they must form a basis of
Rn.
True: n linearly independent vectors in a space of
dimension n form a basis.
4. There is a 5x4 matrix whose image
consists of all of R5.
False. It takes at least 5 vectors to span
all of R5.
5. The kernel of any invertible matrix
consists of the zero vector only.
True. AX = 0 implies X = 0 when A is
invertible.
6. The identity matrix In is similar to all
invertible nxn matrices.
False. The identity matrix is similar only to
itself. A-1 I A = I for all invertible matrices
A.
7. If 2 U + 3 V + 4 W = 5U + 6 V + 7 W, then
vectors U, V, W must be linearly
dependent.
True. In fact 3U+3V+3W = 0.
8. The column vectors of a 5x4 matrix must be
linearly dependent.
False.
|1 0 0 0|
|0 1 0 0|
|0 0 1 0|
|0 0 0 1|
|0 0 0 0|
is an example where they are linearly
independent.
9. If V1, V2, …, Vn and W1, W2, …, Wm are
any two bases of a subspace V of R10,
then n must equal m.
True. Any two bases of the same vector
space have the same number of vectors.
10. If A is a 5x6 matrix of rank 4, then the
nullity of A is 1.
False. The rank plus the nullity is the
number of columns. Thus the nullity would
be 2.
11. If the kernel of a matrix A consists of the
zero vector only, then the column vectors
of A must be linearly independent.
True. Since the kernel is zero, the columns
of A must be linearly independent.
12. If the image of an nxn matrix A is all of
Rn, then A must be invertible.
True. Since the columns span Rn , the
matrix must have a right inverse. Since it
is square, it must be invertible.
13. If vectors V1, V2, …, Vn span R4 then n
must be equal to 4.
False. It could be 4 or larger than 4.
14. If vectors U, V, and W are in a subspace
V of Rn, then 2 U – 3 V + 4 W must be in V
as well.
True. A subspace is closed under addition
and scalar multiplication.
15. If matrix A is similar to matrix B, and B is
similar to C, then C must be similar to A.
True.
P-1AP = B
Q-1BQ = C
Q-1P-1APQ = C
A = PQCQ-1P-1
A = (Q-1P-1)-1 C (Q-1P-1)
16. If a subspace V of Rn contains none of
the standard vectors E1, E2, …, En, then V
consists of the zero vector only.
|c|
False. The space | c | of R3 is a
|c|
counter example.
17. If vectors V1, V2, V3, V4 are linearly
independent, then vectors V1, V2, V3 must
be linearly independent as well.
True. Any dependence relation among V1,
V2, V3 can be made into a dependence
relation for V1, V2, V3, V4 by adding a zero
coefficient to V4.
| a |
18. The vectors of the form | b |
| 0 |
| a |
(where a and b are arbitrary real numbers) form a
subspace of R4.
True. This is closed under addition and scalar
multiplication.
19. Matrix | 1 0 | is similar to | 0 1 |.
| 0 -1 |
|1 0|
-1
True. |1/2 -1/2 | | 1 0| |1/2 -1/2 | = | 0 1 |
|1/2 1/2 | | 0 -1| |1/2 1/2|
|10|
| 1 | |2| |3|
20. Vectors | 0 |, | 1 |, | 2 | form a basis of R3.
| 0 | |0| |1|
| 1 | | 2 | | 3 | |a+2b+3c|
True.
a| 0 |+b| 1 |+c| 2 | = | b+2c |
|0| |0| |1| | c
|
For the dependence relation to equal zero,
we must have c = 0, then b=0, then a=0. Thus
the three vectors are linearly independent and
must be a basis of R3.
21. Matrix | 0 1 | is similar to | 0 0 |.
|0 0|
| 0 1 |
False. The first matrix squares to zero. The
second matrix does not square to zero. They
cannot be similar.
22. These vectors are linearly independent.
|1|
|2|
|3|
|4|
|5|
|6|
|7|
|8|
|9|
|8|
|7|
|6|
|5|
|4|
|3|
|2|
|1|
|0|
|-1 |
|-2 |
False. They are five vectors in a space of
dimension 4. They must be linearly dependent.
23. If a subspace V of R3 contains the standard
vectors E1, E2, E3, then V must be R3.
True. Clearly everything is a linear combination
of E1, E2, and E3.
24. If a 2x2 matrix P represents the orthogonal
projection onto a line in R2, then P must be
similar to matrix | 1 0 |.
| 0 0|
True. Use one basis vector along the line
things are projected onto, and put the other
basis vector along the line perpendicular to the
first.
25. If A and B are nxn matrices, and vector V is
in the kernel of both A and B, then V must be in
the kernel of matrix AB as well.
True. In fact we did not even need V to be in the
kernel of A. If V is in the kernel of B, then V is
in the kernel of AB.
26. If two nonzero vectors are linearly
dependent, then each of them is a scalar
multiple of the other.
True.
The dependence relation aV+bW = 0
has to have both a and b nonzero. Then
V = -b/a W and W = -a/b V.
27. If V1, V2, V3 are any three vectors in R3, then
there must be a linear transformation T from R3
to R3 such that T(V1) = E1, T(V2) = E2, and T(V3)
= E3.
False. You can do this when they are
independent. You cannot do it when they are
dependent.
28. If vectors U, V, W are linearly dependent,
then vector W must be a linear combination of
U and V.
False. Let U = V = 0 and W = E3.
29. If A and B are invertible nxn matrices, then
AB is similar to BA.
True.
A-1(AB)A = BA
30. If A is an invertible nxn matrix, then the
-1
kernels of A and A must be equal.
True. In fact the kernels of A and A-1 are both
just 0.
31. If V is any three-dimensional subspace of R5
then V has infinitely many bases.
True.
If V1, V2, V3 is one basis, then
V1+kV2, V2, V3 is another basis for
each integer k.
32. Matrix In is similar to 2 In.
False. In is similar to only itself.
33. If AB = 0 for two 2x2 matrices A and B, then
BA must be the zero matrix as well.
False.
|0 0|
|1 0|
|0 0| = | 0 0 |
|0 1|
| 0 0 |
|0 0|
|0 1|
|0 0| = | 0 0 |
|1 0|
| 1 0 |
34. If A and B are nxn matrices, and V is in the
image of both A and B, then V must be in the
image of matrix A+B as well.
False. Consider B = -A. Then
A+B = 0 yet A and B have the same image.
35. If V and W are subspaces of Rn, then their
union VuW must be a subspace of Rn as well.
False. V = | c |
|0|
W = | 0 |.
|d|
Then VuW is not closed under addition since
| c | is not in the union.
|d|
36. If the kernel of a 5x4 matrix A consists of the
zero vector only and if AV = AW for two vectors
V and W in R4, then vectors V and W must be
equal.
True. Since A(V-W) = 0, V-W = 0 and so V=W.
37. If V1, V2, …, Vn and W1, W2, …, Wn are two
bases of Rn, then there is a linear
transformation T from Rn to Rn such that T(V1) =
W1, T(V2) = W2, …, T(Vn) = Wn.
True. You can map a basis anywhere.
38. If matrix A represents a rotation through Pi/2
and matrix B rotation through Pi/4, then A is
similar to B.
False.
A = | 0 -1 |
| 1 0|
B = | 1/Sqrt[2] -1/Sqrt[2] |
| 1/Sqrt[2] 1/Sqrt[2] |
A4 = I and B4 =/= I. They cannot be similar.
39. R2 is a subspace of R3.
False. There are subspaces of R3 of dimension
2, but the vectors in them are all three tuples,
not 2 tuples.
40. If an nxn matrix A is similar to matrix B, then
A + 7In must be similar to B + 7 In.
True. If P-1AP = B then
P-1(A+7In)P = P-1AP + 7 P-1 In P = B + 7 In
41. There is a 2x2 matrix A such that im(A) =
ker(A).
True.
| 0 1 | is one such matrix.
|00 |
42. If two nxn matrices A and B have the same
rank, then they must be similar.
False.
| 1 0 | and | 0 1 | both have rank
|0 0|
|00|
one, but are not similar.
43. If A is similar to B, and A is invertible, then B
must be invertible as well.
True.
If P-1 A P = B then
P-1 A-1 P = B-1
44. If A2 = 0 for a 10x10 matrix A, then the
inequality rank(A) <= 5 must hold.
True. 10 = rank(A) + nullity(A)
Since A is contained in the null space of A,
10 >= 2 rank(A). So rank(A) <= 5.
45. For every subspace V of R3 there is a 3x3
matrix A such that V = im(A).
True. Just pick 3 vectors which span V. Use
these as the columns of the matrix.
46. There is a nonzero 2x2 matrix A that is
similar to 2A.
True.
| 2 0| |01| |½ 0| =|0 2|
| 0 1| |00| |0 1| | 0 0|
47. If the 2x2 matrix R represents the reflection
across a line in R2, then R must be similar to
the matrix | 0 1 |.
| 1 0 |
True. Use the basis
| /
_____|/_______
|\
| \
48. If A is similar to B, then there is one and only
-1
one invertible matrix S such that S A S = B.
False.
(A-1S)-1 A (A-1 S) will also work.
49. If the kernel of a 5x4 matrix A consists of the
zero vector alone, and if AB = AC for two 4x5
matrices B and C, then the matrices B and C
must be equal.
True. A(B-C) = 0 so B-C = 0 and so B=C.
50. If A is any nxn matrix such that A2 = A, then
the image of A and the kernel of A have only the
zero vector in common.
True. If A(AV) = 0, that is, if AV is in the image
of A and also in the kernel of A, then
0 = A2V = AV.
51. There is a 2x2 matrix A such that A2 =/= 0
and A3 = 0.
False. If A2 V =/= 0, and A3 V = 0, then
V, AV, A2V must be linearly independent.
This is impossible in R2.