Consider the following system of linear equations: x + 4y + z = 3, z = -1. Then which of the following is true? Select all possible answers. One of the variables is free. It is a homogeneous system of linear equations. |It has exactly one solution. It has a non-trivial solution.
Consider the following system of linear equations: x + 4y + z = 3, z = -1. Then which of the following is true? Select all possible answers. One of the variables is free. It is a homogeneous system of linear equations. |It has exactly one solution. It has a non-trivial solution.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the following system of linear equations:
x + 4y + z = 3,
Z = -1.
Then which of the following is true?
Select all possible answers.
One of the variables is free.
It is a homogeneous system of linear equations.
It has exactly one solution.
It has a non-trivial solution.
{0000-
1
1
1
1
8
Let V
be a subset of R4. Recall that the span of any set of
1
vectors is a subspace. What is the dimension of span(V)?
4
3
1
{ei, e2, e3 }be an ordered basis for the vector space V. Suppose T : V → V is a linear
1, 2and 3. Then the
matrix (with the respect to the given basis B) that corresponds to the linear map T : V → V must
Let B
map such that T(en) is the linear combination of the first n vectors of B for n
be
Se'ect all possible answers.
diagonal
upper triangular
lower triangular
singular
non-singular
Which of the following must be true?
For any two n x nmatrices A and B, we must have AB =BA
For an invertible matrix M, we may find a large natural number n such that M" is NOT invertible.
1
()
If M =
1
then M³ + M² = 2M
There are some non-invertible elementary matrices.
()
()
1
Suppose
2
and
are the first two columns of a matrix A. If the RREF of the matrix A is
3
1
1
then the third column of the matrix A is
0 0
3
3
2
There could be infinitely many possibilities for the third column of A.
![Let V be the set of all (5x5) real symmetric matrices. Recall that a matrix is called symmetric if it is
equal to its transpose. It can be shown that V is a vector space over R with respect to usual matrix
addition and scalar multiplication. What is the dimension of the vector space V?
10
15
25
Suppose M is a (11x11) matrix such that the dimension of its column space is 10 .Then the rank of M
is
Select all possible answers.
11
10
Let V be a 6 dimensional vector space over the set of reals. Then the dimension of a subspace of V
could be
Select all possible answers.
5
6.
|4
|3
Recall that for any two given real vector spaces V and W, a function T : V → W is linear if T(ax+by) = aT(x)+ bT(y) for any vector x, y in V and for any
real numbers a and b. Let T: R → R be a linear transformation such that
T
= T
1
= T
1
Suppose
10
а
T
1
8
Write down the value of a:
Write down the value of b:
Write down the value of c:
Let M be a (7x7) matrix such that the dimension of its row space is 6. Which of the following is true?
Select all possible answers.
rank of Mequals to 6.
Mis invertible.
Mis not invertible.
dim(columnspace(M))
6.
Let A be a (11x11) matrix such that the dimension of its column space is 11. Then
Select all possible answers.
dim (rowspace (A)) = 11
Ais invertible.
The determinant of Ais a non-zero number.
A-' does not exist.
Let A be a 3 x 3matrix with rank 3. Suppose there are two 3 × 3matrices Land U such that A = LU
. Then which of the following is true?
Select all possible answers.
|rank(A) = rank(L)
|rank(A) = rank(U).
Both U and L are invertible.
column space(A) = column space(L)
L+Umust be invertible.
Let A be the following matrix
4 4 4
6.
6
Then for which column vector b, the equation Ax
bhas a unique solution?
Select all possible answers.
4
6
6
а
The matrix U
0 d
must have linearly dependent columns if
e
0 0
f
]f = 0
|b =c=0
e = 0
a = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf178725-6773-40ca-97b6-519e6c5bce4d%2F1b059e08-9373-470d-841d-326b5d969bf5%2F4p4fqfo_processed.png&w=3840&q=75)
Transcribed Image Text:Let V be the set of all (5x5) real symmetric matrices. Recall that a matrix is called symmetric if it is
equal to its transpose. It can be shown that V is a vector space over R with respect to usual matrix
addition and scalar multiplication. What is the dimension of the vector space V?
10
15
25
Suppose M is a (11x11) matrix such that the dimension of its column space is 10 .Then the rank of M
is
Select all possible answers.
11
10
Let V be a 6 dimensional vector space over the set of reals. Then the dimension of a subspace of V
could be
Select all possible answers.
5
6.
|4
|3
Recall that for any two given real vector spaces V and W, a function T : V → W is linear if T(ax+by) = aT(x)+ bT(y) for any vector x, y in V and for any
real numbers a and b. Let T: R → R be a linear transformation such that
T
= T
1
= T
1
Suppose
10
а
T
1
8
Write down the value of a:
Write down the value of b:
Write down the value of c:
Let M be a (7x7) matrix such that the dimension of its row space is 6. Which of the following is true?
Select all possible answers.
rank of Mequals to 6.
Mis invertible.
Mis not invertible.
dim(columnspace(M))
6.
Let A be a (11x11) matrix such that the dimension of its column space is 11. Then
Select all possible answers.
dim (rowspace (A)) = 11
Ais invertible.
The determinant of Ais a non-zero number.
A-' does not exist.
Let A be a 3 x 3matrix with rank 3. Suppose there are two 3 × 3matrices Land U such that A = LU
. Then which of the following is true?
Select all possible answers.
|rank(A) = rank(L)
|rank(A) = rank(U).
Both U and L are invertible.
column space(A) = column space(L)
L+Umust be invertible.
Let A be the following matrix
4 4 4
6.
6
Then for which column vector b, the equation Ax
bhas a unique solution?
Select all possible answers.
4
6
6
а
The matrix U
0 d
must have linearly dependent columns if
e
0 0
f
]f = 0
|b =c=0
e = 0
a = 0
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