d) Let x = (1, 2, 3) C3 be arbitrary and let a = (a1, 02, 03) € C³, b= (b₁,b2, b) C³ be fixed. Let a1 a2 a3 TH-4639 T(x)= det b₁ b₂ b3 22 23/ Find T(303). ii) Use the symbol T(x) to explain what it means that the determinant of 3 x 3 matrix is a linear map with respect to the third row. iii) Use the result in the preceding part, or otherwise, to prove that T(x) = 0, if x E span {a, b}. Hint: You can use without proof the fact that the determinant of a 3 x 3 matrix with two identical rows is zero.
d) Let x = (1, 2, 3) C3 be arbitrary and let a = (a1, 02, 03) € C³, b= (b₁,b2, b) C³ be fixed. Let a1 a2 a3 TH-4639 T(x)= det b₁ b₂ b3 22 23/ Find T(303). ii) Use the symbol T(x) to explain what it means that the determinant of 3 x 3 matrix is a linear map with respect to the third row. iii) Use the result in the preceding part, or otherwise, to prove that T(x) = 0, if x E span {a, b}. Hint: You can use without proof the fact that the determinant of a 3 x 3 matrix with two identical rows is zero.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.3: Properties Of Determinants
Problem 63E: Let A be an nn matrix in which the entries of each row sum to zero. Find |A|.
Related questions
Question
Only Part D needed
Needed to be solved Part A Correctly in 15 minutes and get the thumbs up please show neat and clean work for it
![1. a)
b)
c)
DL
ii
i)
ii)
iii)
iv)
Use the maple output at the end of this question to give answers to the following
questions:
1-
i)
ii)
3] Let b (b1,b2, b3, b₁) € R4 and
(-
col(A).
ii)
Suppose that A € Mkn (R). Define the following terms:
the column space of the matrix A, col(A) CR.
the nullspace of the matrix A, null (A) CR"
{
null (A).
col(A)?
PRA
Use the maple output
questions:
7-
A =
s] Find the basis and the dimension of the column space of the matrix A,
3] Find the basis and the dimension of the nullspace of the matrix A,
:] What are the conditions on the components of the vector b such that be
s] Find the coordinate vector of the quadruple c = (1, 3, -2, 3)¹ € col (4)
with respect to the basis you gave in response to part i) above.
Let
-1
1 3
3
-3 -3
0
0
-2
-1 1 5
of 3 x 3
B =
1
1 2
--0--0
and y=
1 3
4.
REY
-] Find the projection of the vector y onto the subspace col (B).
s] Let Q be the matrix in the maple output (4) at the end of this question.
Explain why QQT = B(BTB) ¹BT.
•
iii).
best fits the points (1, 1), (2,0),
Explain why the line y = - +
(3, 2), (4,5) in the least squares sense.
Let x = (1, 22, 23)
b= (b₁,b2, b) € C³ be fixed. Let
C³ be arbitrary and let a = (a₁, 02, 03) € C³,
2
at the end of this question to give answers to the following
T(x) = det
a₁ a2 a3
b1 b2 b3
x1 12 13/
Find T(3e3).
Use the symbol T(x) to explain what it means that the determinant
matrix is a linear map with respect to the third row.
Use the result in the preceding part, or otherwise, to prove that T(x) =
0, if x span {a, b}. Hint: You can use without proof the fact that the determinant
of a 3 x 3 matrix with two identical rows is zero.
INSW](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2e672a1-54a8-4743-aad6-f3d1c4a0c4ab%2Fe9905663-0d94-475f-9267-970b2b2cc41b%2Fdwvrq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. a)
b)
c)
DL
ii
i)
ii)
iii)
iv)
Use the maple output at the end of this question to give answers to the following
questions:
1-
i)
ii)
3] Let b (b1,b2, b3, b₁) € R4 and
(-
col(A).
ii)
Suppose that A € Mkn (R). Define the following terms:
the column space of the matrix A, col(A) CR.
the nullspace of the matrix A, null (A) CR"
{
null (A).
col(A)?
PRA
Use the maple output
questions:
7-
A =
s] Find the basis and the dimension of the column space of the matrix A,
3] Find the basis and the dimension of the nullspace of the matrix A,
:] What are the conditions on the components of the vector b such that be
s] Find the coordinate vector of the quadruple c = (1, 3, -2, 3)¹ € col (4)
with respect to the basis you gave in response to part i) above.
Let
-1
1 3
3
-3 -3
0
0
-2
-1 1 5
of 3 x 3
B =
1
1 2
--0--0
and y=
1 3
4.
REY
-] Find the projection of the vector y onto the subspace col (B).
s] Let Q be the matrix in the maple output (4) at the end of this question.
Explain why QQT = B(BTB) ¹BT.
•
iii).
best fits the points (1, 1), (2,0),
Explain why the line y = - +
(3, 2), (4,5) in the least squares sense.
Let x = (1, 22, 23)
b= (b₁,b2, b) € C³ be fixed. Let
C³ be arbitrary and let a = (a₁, 02, 03) € C³,
2
at the end of this question to give answers to the following
T(x) = det
a₁ a2 a3
b1 b2 b3
x1 12 13/
Find T(3e3).
Use the symbol T(x) to explain what it means that the determinant
matrix is a linear map with respect to the third row.
Use the result in the preceding part, or otherwise, to prove that T(x) =
0, if x span {a, b}. Hint: You can use without proof the fact that the determinant
of a 3 x 3 matrix with two identical rows is zero.
INSW
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