1. Consider Let R(A) column space of A R(A) = column space of AT, 0 1210 2511 0 372 2 493-14 by N(A) = null space of A N(AT) = null space of AT. (a) By finding the linearly independent columns, express the four fundamental spaces: R(A), R(AT), N(A), and N(AT). Also find their dimensions. b₂ (b) Find all the vectors b by b₁ [Try to establish a relation in the components of b.] Using this result, decide for -2 what value of kER the system Ax=b is inconsistent where b = [Instruction: Do not solve Ax=b explicitly.] (e) Do you see any relation between the vector b for which Ax=b is consistent and N(AT)? Can you prove this relation in general? ER such that the system Ax=b is consistent.
1. Consider Let R(A) column space of A R(A) = column space of AT, 0 1210 2511 0 372 2 493-14 by N(A) = null space of A N(AT) = null space of AT. (a) By finding the linearly independent columns, express the four fundamental spaces: R(A), R(AT), N(A), and N(AT). Also find their dimensions. b₂ (b) Find all the vectors b by b₁ [Try to establish a relation in the components of b.] Using this result, decide for -2 what value of kER the system Ax=b is inconsistent where b = [Instruction: Do not solve Ax=b explicitly.] (e) Do you see any relation between the vector b for which Ax=b is consistent and N(AT)? Can you prove this relation in general? ER such that the system Ax=b is consistent.
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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Question
![1. Consider
2. Consider the following vector v E R' and the subspace SC R4.
1 2 1 0
2 5 1 1
3 7 2 2
4 9 3 -1
A =
-
-2
4
v =
3.
S = span
Let
R(A) = column space of A
N(A) = null space of A,
N(A") = null space of A".
R(A") = column space of A",
(a) Find an orthonormal basis B for the subspace S.
(a) By finding the linearly independent columns, express the four fundamental spaces:
R(A), R(A"), N(A), and N(A"). Also find their dimensions.
(b) Find the projection of v onto the subspace S. Also find the coordinate matrix
[projsv]B.
(c) Your friend wants to calculate the coordinate matrix of an arbitrary vector u € R
relative to the orthonormal basis B but she is not being able to do it. Can you help
her? Which method would you suggest her? Explain why do you think the method
would give her the desired result?
bị
b2
E R' such that the system Ax = b is consistent.
b3
(b) Find all the vectors b =
[Try to establish a relation in the components of b.] Using this result, decide for
-2
3. Consider that matrix
what value of k ER the system Ax = b is inconsistent where b =
1 2
2 4
-1
A =
[Instruction: Do not solve Ax = b explicitly.]
(a) Find the eigenvalues of A. Also find the corresponding eigenspaces and their
dimensions.
(c) Do you see any relation between the vector b for which Ax = b is consistent and
N(A")? Can you
this relation in general?
(b) Explain why A is diagonalizable. Calculate A2021.
(c) Your friend claims
“I think if P diagonalizes A, then P diagonalizes every integral power of A."
Is his statement correct? If not, under what condition(s) the statement holds true?
Can you prove it in general?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd2598f8-0d4c-48b5-abd4-a817c00d8903%2F967e0b0e-c210-4933-8154-d2a588962b20%2Fagofd9_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider
2. Consider the following vector v E R' and the subspace SC R4.
1 2 1 0
2 5 1 1
3 7 2 2
4 9 3 -1
A =
-
-2
4
v =
3.
S = span
Let
R(A) = column space of A
N(A) = null space of A,
N(A") = null space of A".
R(A") = column space of A",
(a) Find an orthonormal basis B for the subspace S.
(a) By finding the linearly independent columns, express the four fundamental spaces:
R(A), R(A"), N(A), and N(A"). Also find their dimensions.
(b) Find the projection of v onto the subspace S. Also find the coordinate matrix
[projsv]B.
(c) Your friend wants to calculate the coordinate matrix of an arbitrary vector u € R
relative to the orthonormal basis B but she is not being able to do it. Can you help
her? Which method would you suggest her? Explain why do you think the method
would give her the desired result?
bị
b2
E R' such that the system Ax = b is consistent.
b3
(b) Find all the vectors b =
[Try to establish a relation in the components of b.] Using this result, decide for
-2
3. Consider that matrix
what value of k ER the system Ax = b is inconsistent where b =
1 2
2 4
-1
A =
[Instruction: Do not solve Ax = b explicitly.]
(a) Find the eigenvalues of A. Also find the corresponding eigenspaces and their
dimensions.
(c) Do you see any relation between the vector b for which Ax = b is consistent and
N(A")? Can you
this relation in general?
(b) Explain why A is diagonalizable. Calculate A2021.
(c) Your friend claims
“I think if P diagonalizes A, then P diagonalizes every integral power of A."
Is his statement correct? If not, under what condition(s) the statement holds true?
Can you prove it in general?
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